1.
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!
2.
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.
3.
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.
4.
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
5.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
6.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
7.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
8.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
9.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
10.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
11.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
12.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
13.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
14.
From the sequence above, can an infinite geometric series be calculated?
Correct Answer
C. No, it's not even geometric
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.
15.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
16.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
17.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
18.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
19.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
20.
From the sum of the sequence above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
21.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
22.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
23.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
24.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
25.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
B. No, it is geometric but it diverges
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.
26.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
27.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
28.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
29.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
30.
From the values given above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
31.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
32.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
33.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
34.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
35.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
36.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
37.
From the values given in the series above, can an infinite geometric series be calculated?
Correct Answer
A. Yes, it is geometric and it is converging
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.
38.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
4096
39.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
5/4
1.25
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.
40.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
5.2
41.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
4
Explanation
The value of the infinite series is 4.
42.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
6.25
43.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
54
44.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
2.2
45.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
5
Explanation
The value of the infinite series is 5.
46.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
-16
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.
47.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
27
48.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
none
49.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
-12
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.
50.
Given the values above, what is the value of the infinite series? (It there is no infinite series, type 'none'.)
Correct Answer
48