9046 Number Patterns Geometric Infinite Series

  • Grade 7th,
  • Grade 8th,
  • Grade 9th,
  • Grade 10th
  • CCSS.MATH.CONTENT.HSA.SSE.B.4
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| Attempts: 204 | Questions: 95 | Updated: Mar 20, 2025
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1) I spill a 10 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 100 square centimetres. In the first minute it has spread by 30 square centimetres, and is growing by 50% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit
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About This Quiz
Geometric Sequences Quizzes & Trivia

Explore the application of geometric series in real-world scenarios through this engaging quiz titled '9046 Number Patterns Geometric Infinite Series'. Assess your understanding of mathematical concepts as you solve problems involving bouncing superballs, bounding kangaroos, spreading paint stains, and skipping stones.

2) I'm standing by a lake, skipping stones. My first skip is 6 metres, and each successive skip is 40% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

3) I'm standing by a lake, skipping stones. My first skip is 6 metres, and each successive skip is 50% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

4) I'm standing by a lake, skipping stones. My first skip is 6 metres, and each successive skip is 20% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

5) I'm standing by a lake, skipping stones. My first skip is 6 metres, and each successive skip is 30% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

6) I'm standing by a lake, skipping stones. My first skip is 4 metres, and each successive skip is 40% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

7) I'm standing by a lake, skipping stones. My first skip is 4 metres, and each successive skip is 30% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

8) I'm standing by a lake, skipping stones. My first skip is 4 metres, and each successive skip is 20% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

9) I spill a 20 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 100 square centimetres. In the first minute it has spread by 60 square centimetres, and is growing by 50% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

10) I spill a 20 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 100 square centimetres. In the first minute it has spread by 40 square centimetres, and is growing by 20% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

11) I spill a 20 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 160 square centimetres. In the first minute it has spread by 40 square centimetres, and is growing by 20% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

12) I spill a 20 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 160 square centimetres. In the first minute it has spread by 50 square centimetres, and is growing by 50% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

13) I spill a 10 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 80 square centimetres. In the first minute it has spread by 60 square centimetres, and is growing by 50% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

14) I spill a 10 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 150 square centimetres. In the first minute it has spread by 40 square centimetres, and is growing by 50% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

15) I spill a 10 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 100 square centimetres. In the first minute it has spread by 30 square centimetres, and is growing by 20% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

Submit

16) A baby kangaroo is grazing in a paddock 20 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 20% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The kangaroo's first leap is 5 meters. Each successive leap is 20% less than the previous leap. So, the second leap would be 80% of 5 meters, which is 4 meters. The third leap would be 80% of 4 meters, which is 3.2 meters. This pattern continues until the kangaroo reaches safety. By calculating the successive leaps, we find that the kangaroo will take 8 leaps to reach safety.

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17) A baby kangaroo is grazing in a paddock 15 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 30% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The baby kangaroo takes a leap of 5 meters in the first jump. Each subsequent leap is 30% less than the previous leap. This means that the distances of the leaps form a decreasing geometric sequence. To find the number of leaps it will take for the kangaroo to reach safety, we need to determine when the sum of the distances of the leaps will be greater than or equal to 15 meters. By calculating the sum of the geometric sequence, we find that it will take 7 leaps for the kangaroo to reach safety.

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18) A baby kangaroo is grazing in a paddock 40 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 10% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The kangaroo's first leap is 5 meters. Each successive leap is 10% less than the previous leap. This means that each leap is 90% of the previous leap. To find the number of leaps it will take for the kangaroo to reach safety, we can divide the distance to the fence (40 meters) by the distance covered in each leap. Since each leap is 90% of the previous leap, the distance covered in each leap can be calculated by multiplying the previous leap distance by 0.9. By repeating this calculation for each leap, we find that it will take 16 leaps for the kangaroo to reach safety.

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19) A baby kangaroo is grazing in a paddock 35 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 10% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The baby kangaroo takes a leap of 5 meters for the first jump. Each successive leap is 10% less than the previous leap. This means that for each leap, the distance covered will be 90% of the previous distance. To calculate the number of leaps needed to reach safety, we can divide the initial distance of 35 meters by the distance covered in each leap. The distance covered in each leap can be calculated by multiplying the previous distance by 0.9. By repeating this process, we find that it will take 12 leaps for the kangaroo to reach safety.

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20) A baby kangaroo is grazing in a paddock 30 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 10% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The baby kangaroo takes a leap of 5 meters for the first jump. Each successive leap is 10% less than the previous leap, so the distances for the next leaps would be 4.5 meters, 4.05 meters, 3.645 meters, and so on. To find the number of leaps it will take to reach safety, we need to determine when the total distance covered by the leaps exceeds or equals 30 meters. By calculating the sum of the distances covered by the first 9 leaps (5 + 4.5 + 4.05 + ...), we can see that it will take 9 leaps for the kangaroo to reach safety.

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21) A baby kangaroo is grazing in a paddock 26 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 10% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The kangaroo takes 5 meters for the first leap. Each successive leap is 10% less than the previous leap, so the distances for the next leaps are 4.5 meters, 4.05 meters, 3.645 meters, 3.2805 meters, 2.95245 meters, and 2.657205 meters. The total distance covered by the kangaroo after these 7 leaps is approximately 26 meters, which is the distance to reach safety. Therefore, it will take the kangaroo 7 leaps to reach safety.

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22) A baby kangaroo is grazing in a paddock 22 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 10% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The kangaroo takes its first leap of 5 metres. Each successive leap is 10% less than the previous leap, so the second leap is 4.5 metres, the third leap is 4.05 metres, the fourth leap is 3.645 metres, the fifth leap is 3.2805 metres, and the sixth leap is 2.95245 metres. In total, the kangaroo takes 6 leaps to reach safety.

Submit

23) A baby kangaroo is grazing in a paddock 22 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 20% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The baby kangaroo takes a leap of 5 meters, and each successive leap is 20% less than the previous leap. This means that each leap is 80% of the previous leap. To calculate the distance covered in each leap, we multiply the distance of the previous leap by 0.8.

The distances covered in each leap are as follows:
1st leap: 5 meters
2nd leap: 5 * 0.8 = 4 meters
3rd leap: 4 * 0.8 = 3.2 meters
4th leap: 3.2 * 0.8 = 2.56 meters
5th leap: 2.56 * 0.8 = 2.048 meters
6th leap: 2.048 * 0.8 = 1.6384 meters
7th leap: 1.6384 * 0.8 = 1.31072 meters
8th leap: 1.31072 * 0.8 = 1.048576 meters
9th leap: 1.048576 * 0.8 = 0.8388608 meters
10th leap: 0.8388608 * 0.8 = 0.67108864 meters

Since the distance covered in the 10th leap is less than 1 meter, the kangaroo will not be able to reach safety in the 10th leap. Therefore, it will take the kangaroo 10 leaps to reach safety.

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24) A superball is dropped from a building 12 metres high. It each bounce, it loses 50% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

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25) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The value of the infinite series is 5.

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26) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is both geometric and converging. A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. In this case, the series satisfies this condition. Additionally, the fact that it is converging means that the sum of the terms in the series approaches a finite value as the number of terms increases. Therefore, an infinite geometric series can be calculated from the given values.

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27) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The value of r is 0.2 because it is given in the question that the first term (a1) is 0.2 and the sum of the sequence (S) is also 0.2. In a geometric sequence, the sum of the sequence is calculated using the formula S = a1 * (1 - r^n) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms. Since S is equal to a1 in this case, we can substitute the values and solve for r, which gives us r = 0.2.

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28) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is geometric because each term is obtained by multiplying the previous term by a constant ratio. Additionally, it is converging because the terms in the series are getting smaller and approaching a finite limit as the series progresses. Hence, an infinite geometric series can be calculated.

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29) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The value of r is 0.9 because it is given that the first term (a1) is 0.9 and the sum of the sequence (S) is also 0.9. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio (r). Since the sum of the sequence is equal to the first term in this case, it implies that there is only one term in the sequence, and that term is equal to the first term. Therefore, the value of r must be 0.9.

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30) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is geometric because each term is obtained by multiplying the previous term by a constant ratio. Additionally, the series is converging because the terms are getting smaller and approaching a finite limit as the series progresses. Therefore, an infinite geometric series can be calculated.

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31) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is both geometric and converging. A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. In this case, the series follows this pattern, making it geometric. Additionally, the series is converging, which means that as more terms are added, the sum of the series approaches a finite value. Therefore, an infinite geometric series can be calculated from the values given.

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32) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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33) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The given answer is correct because both 5/4 and 1.25 represent the same value. 5/4 is a fraction that can be simplified to 1.25 in decimal form. Therefore, both expressions are equivalent and represent the value of the infinite series.

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34) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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35) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The value of the infinite series is 4.

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36) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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37) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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38) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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39) From the values given above, can an infinite geometric series be calculated?

Explanation

The given answer suggests that an infinite geometric series can be calculated because it meets the criteria of being geometric and converging. A geometric series is one in which each term is found by multiplying the previous term by a constant ratio. In this case, the series is both geometric and converging, meaning that the terms are related by a constant ratio and the series approaches a finite limit as the number of terms increases. Therefore, it is possible to calculate an infinite geometric series.

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40) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The question asks for the value of an infinite series, but the answer is -16. This suggests that there is no infinite series and the value is simply -16.

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41) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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42) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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43) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The answer is -12 because the question asks for the value of the infinite series, and the given value is -12. Therefore, the value of the infinite series is -12.

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44) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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45) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The given answer is -3. However, without any context or additional information, it is not possible to determine the value of the infinite series. The question is incomplete and lacks the necessary details to provide a specific explanation.

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46) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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47) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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48) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

The given question asks for the value of an infinite series. However, the answer is -20, which means that there is an infinite series and its value is -20.

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49) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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50) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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51) Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)

Explanation

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52) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The given answer is correct because both 1/2 and 0.5 represent the same value. They are both equivalent decimal representations of the fraction 1/2. Therefore, either 1/2 or 0.5 can be the value of r in this case.

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53) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

The sequence mentioned in the question is geometric, meaning that each term is obtained by multiplying the previous term by a constant ratio. However, the fact that it diverges means that the terms of the sequence do not approach a finite value as the number of terms increases. Therefore, an infinite geometric series cannot be calculated from this sequence.

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54) A baby kangaroo is grazing in a paddock 20 metres outside the National Park. He sees a ute full of noisy teenagers driving his way, and decides that the park might be safer than a farmers paddock. He bounds toward the fence, his first leap taking him 5 metres. Each successive leap is 20% less than the previous leap. How many leaps will it take the kangaroo to reach safety?

Explanation

The baby kangaroo takes a 5-meter leap initially. Each subsequent leap is 20% less than the previous leap. This means that the length of each leap can be calculated by multiplying the previous leap by 0.8.

So, the lengths of the leaps would be:
1st leap: 5 meters
2nd leap: 5 * 0.8 = 4 meters
3rd leap: 4 * 0.8 = 3.2 meters
4th leap: 3.2 * 0.8 = 2.56 meters
5th leap: 2.56 * 0.8 = 2.048 meters
6th leap: 2.048 * 0.8 = 1.6384 meters
7th leap: 1.6384 * 0.8 = 1.31072 meters
8th leap: 1.31072 * 0.8 = 1.048576 meters

Therefore, it will take the kangaroo 8 leaps to reach safety.

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55) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

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56) I'm standing by a lake, skipping stones. My first skip is 5 metres, and each successive skip is 30% less. What is the total distance (to the nearest whole number) my stone will travel before it stops skipping?

Explanation

Use ar, where R is converted to r. Find the infinite series

Submit

57) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is not even geometric because a geometric sequence is one in which each term is obtained by multiplying the previous term by a constant. In this case, there is no common ratio between the terms, so it cannot be considered a geometric series.

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58) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The value of r is 0.8 because it is given that the first term (a1) is 0.8 and the sum of the sequence (S) is also 0.8. In a geometric sequence, the sum of the sequence is calculated using the formula S = a1 * (1 - r^n) / (1 - r), where a1 is the first term and r is the common ratio. Since S and a1 are both 0.8, we can substitute these values into the formula and solve for r.

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59) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is not even geometric because a geometric sequence is one in which each term is obtained by multiplying the previous term by a constant ratio. In this sequence, there is no constant ratio between the terms, so it cannot be considered geometric. Therefore, an infinite geometric series cannot be calculated from this sequence.

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60) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is not even geometric because a geometric sequence is one in which each term is obtained by multiplying the previous term by a constant ratio. However, in the given sequence, there is no constant ratio between the terms. Therefore, an infinite geometric series cannot be calculated from this sequence.

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61) From the sequence above, can an infinite geometric series be calculated?

Explanation

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62) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence cannot be considered geometric because a geometric sequence must have a common ratio between consecutive terms. In this case, there is no common ratio between the terms, so it cannot be considered geometric. Therefore, an infinite geometric series cannot be calculated.

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63) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The given answer is 1/2,0.5. This is because both 1/2 and 0.5 represent the same decimal value, which is the value of r. In a geometric sequence, the common ratio (r) is the value that each term is multiplied by to get the next term. In this case, both 1/2 and 0.5 can be multiplied by to get the next term, making them equivalent.

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64) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence does not have a common ratio between its terms, which is a requirement for a sequence to be geometric. Therefore, it cannot be considered as a geometric series.

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65) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

The question is asking whether an infinite geometric series can be calculated from the given sequence. The answer is no, even though the sequence is geometric, it diverges. This means that the terms in the sequence do not approach a finite value as the sequence progresses, making it impossible to calculate the sum of an infinite geometric series.

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66) From the values given above, can an infinite geometric series be calculated?

Explanation

The given answer states that an infinite geometric series can be calculated because it is both geometric and converging. A geometric series is a sequence of numbers in which each term is found by multiplying the previous term by a constant ratio. If the ratio is between -1 and 1, the series is converging, meaning it approaches a finite value as the number of terms increases. Therefore, based on the information provided, it can be concluded that an infinite geometric series can be calculated.

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67) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The given answer choices, 1/5 and 0.2, both represent the value of r in the geometric sequence. This can be determined by using the formula for the sum of a geometric sequence, which is S = a1 * (1 - r^n) / (1 - r), where S is the sum of the sequence, a1 is the first term, r is the common ratio, and n is the number of terms. By substituting the given values into the formula, it can be solved for r.

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68) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The given answer, 1/2 and 0.5, represents the value of r in the geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant ratio, denoted as r. Since 1/2 and 0.5 are equivalent values, they both represent the ratio by which each term is multiplied to obtain the next term in the sequence. Therefore, either 1/2 or 0.5 can be the value of r.

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69) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

The given answer options, 1/5 and 0.2, both represent the same decimal value. Since the question asks for the value of r, which is the common ratio in a geometric sequence, it can be concluded that the common ratio is equal to 1/5 or 0.2.

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70) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is geometric because each term is obtained by multiplying the previous term by a constant ratio. However, the sequence diverges because the terms do not approach a finite value as the number of terms increases. Therefore, an infinite geometric series cannot be calculated from this sequence.

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71) From the values given above, can an infinite geometric series be calculated?

Explanation

The correct answer is No, it is geometric but it diverges. This means that although the series is geometric, it does not converge to a specific value. In a geometric series, each term is obtained by multiplying the previous term by a constant ratio. However, in this case, the terms do not approach a finite value as the series progresses, indicating that it diverges.

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72) From the values given above, can an infinite geometric series be calculated?

Explanation

The given answer states that the series is geometric but it diverges. This means that although the series follows a geometric pattern, it does not have a finite sum and therefore cannot be calculated as an infinite geometric series.

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73) From the values given above, can an infinite geometric series be calculated?

Explanation

The given series is described as geometric, which means that each term is obtained by multiplying the previous term by a constant ratio. However, the fact that it diverges indicates that the series does not have a finite sum. In other words, as the series progresses, the terms become larger and larger, rather than approaching a specific value. Therefore, it is not possible to calculate an infinite geometric series in this case.

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74) From the values given above, can an infinite geometric series be calculated?

Explanation

The given series is described as geometric, meaning that each term is obtained by multiplying the previous term by a constant ratio. However, the series is stated to diverge, which means that it does not have a finite sum. This implies that even though the series follows a geometric pattern, it does not converge to a specific value and therefore cannot be calculated as an infinite geometric series.

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75) From the values given above, can an infinite geometric series be calculated?

Explanation

The given answer suggests that an infinite geometric series can be calculated because it is both geometric and converging. A geometric series is a series in which each term is found by multiplying the previous term by a constant ratio. If the ratio is between -1 and 1, the series converges, meaning it approaches a finite value as the number of terms increases. Therefore, if the series is both geometric and converging, it is possible to calculate an infinite sum.

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76) Given the values above for the first term (a1), and the sum of the sequence (S), what is the value of r?

Explanation

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77) From the values given above, can an infinite geometric series be calculated?

Explanation

The given answer states that an infinite geometric series can be calculated because it is both geometric and converging. A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. If the series is converging, it means that the terms are approaching a finite limit as the number of terms increases. Therefore, it is possible to calculate the sum of the infinite geometric series.

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78) From the values given above, can an infinite geometric series be calculated?

Explanation

The given series is geometric because it follows a pattern where each term is multiplied by a common ratio to get the next term. However, the series diverges because the common ratio is greater than 1. In a converging geometric series, the absolute value of the common ratio should be less than 1. Since the series diverges, it cannot be calculated infinitely.

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79) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is not even geometric because a geometric sequence must have a common ratio between the terms. Without a common ratio, it is not possible to calculate an infinite geometric series.

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80) From the sequence above, can an infinite geometric series be calculated?

Explanation

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81) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is stated to be geometric, which means that each term is obtained by multiplying the previous term by a constant ratio. Additionally, it is mentioned that the series is converging. This implies that the terms of the series are approaching a finite limit as the number of terms increases. Therefore, it is possible to calculate an infinite geometric series using the given values.

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82) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is stated to be geometric and converging. A geometric series is one in which each term is obtained by multiplying the previous term by a constant. If the series is converging, it means that the terms are approaching a finite value as the number of terms increases. Therefore, it is possible to calculate an infinite geometric series in this case.

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83) From the values given in the series above, can an infinite geometric series be calculated?

Explanation

The given series is geometric because it follows a pattern where each term is obtained by multiplying the previous term by a constant ratio. Additionally, it is converging because the terms in the series are approaching a finite limit as the number of terms increases. Therefore, an infinite geometric series can be calculated from the given values.

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84) From the values given above, can an infinite geometric series be calculated?

Explanation

The answer is "Yes, it is geometric and it is converging". This means that the given series can be calculated as an infinite geometric series. A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. If the ratio between consecutive terms is between -1 and 1, then the series is converging, meaning that it approaches a finite limit as the number of terms increases. Since the given series is both geometric and converging, it can be calculated as an infinite geometric series.

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85) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is geometric because each term is obtained by multiplying the previous term by a common ratio. However, it diverges because the common ratio is greater than 1, causing the terms to increase without bound. In an infinite geometric series, the terms must approach zero for the series to converge. Since this sequence does not meet that condition, it cannot be used to calculate an infinite geometric series.

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86) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

A geometric series can only be calculated if the common ratio between consecutive terms is between -1 and 1. In this case, since it is mentioned that the sequence is diverging, it means that the terms are not approaching a specific value and therefore the common ratio is not within the acceptable range. Hence, an infinite geometric series cannot be calculated from this sequence.

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87) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence cannot be considered as a geometric series because a geometric series must have a common ratio between consecutive terms. In this case, there is no common ratio between the terms, so it cannot be considered geometric.

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88) From the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence cannot be considered a geometric series because a geometric series must have a common ratio between each term. In the given sequence, there is no consistent ratio between the terms, so it cannot be considered geometric. Therefore, an infinite geometric series cannot be calculated.

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89) From the sum of the sequence above, can an infinite geometric series be calculated?

Explanation

The given sequence is geometric because there is a common ratio between each term. However, it diverges because the terms of the sequence do not approach a finite value as the number of terms increases. In other words, the sum of the sequence does not converge to a specific value, making it impossible to calculate an infinite geometric series from it.

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90) A superball is dropped from a building 30 metres high. It each bounce, it loses 30% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

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91) I spill a 20 litre can of paint on the carpet. I'm in serious trouble. When it starts, it covers 200 square centimetres. In the first minute it has spread by 40 square centimetres, and is growing by 20% each minute. What is the final area of the stain in square centimetres?

Explanation

Use ar to find the sequence. If it is growing by 20%, the value for r will be 0.2. Find the value for the infinite series. Add to original coverage.

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92) A superball is dropped from a building 20 metres high. It each bounce, it loses 40% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

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93) A superball is dropped from a building 40 metres high. It each bounce, it loses 50% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

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94) A superball is dropped from a building 32 metres high. It each bounce, it loses 50% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

Submit

95) A superball is dropped from a building 26 metres high. It each bounce, it loses 50% of its height. What will be the total distance traveled by the ball until it rests on the ground?

Explanation

Use ar to find the sequence - to see how high it bounces each time. Make sure you convert R to r.
Find the height for the second bounce and double it - it has to come down. Find the bounce for the third bounce and double it. Find the fourth bounce - double that. Use first3 to find the total of the infinite series - then - make sure you add on the distance of the first drop!

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