9900a Arithmetic & Geometric Seq & Series Test A

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The sequence is formed by dividing each term by 2. So, starting with 50, we divide it by 2 to get 25, then divide 25 by 2 to get 12.5, and so on. Continuing this pattern, the 5th term would be obtained by dividing the previous term, 6.25, by 2, which gives us 3.125.

The expression for the nth term in the given sequence can be found by subtracting 3 from each term. This is evident as each term is 3 less than the previous term. Therefore, option B is the correct answer.

In this sequence, each term is obtained by adding 15 to the previous term. So, the second term would be 100 + 15 = 115, the third term would be 115 + 15 = 130, the fourth term would be 130 + 15 = 145, and the fifth term would be 145 + 15 = 160. Therefore, the fifth term in the sequence is 160.

The common ratio of the geometric sequence can be found by dividing any term in the sequence by its previous term. In this case, dividing 2187 by 19683 gives a result of 1/9. Similarly, dividing 243 by 2187 also gives a result of 1/9. Therefore, the common ratio of the geometric sequence is 1/9.

The common ratio for a geometric sequence is found by taking the ratio of any term to its previous term. In this case, since the successive terms increase in value by 4.5%, the ratio of any term to its previous term will be 1 + 0.045, which simplifies to 1.045. Therefore, the common ratio for this geometric sequence is 1.045.

The expression for the nth term of a geometric sequence is given by the formula an = a * r^(n-1), where a is the first term and r is the common ratio. In this case, the first term (a) is 10 and the common ratio (r) is 0.3. Therefore, the expression for the nth term would be an = 10 * 0.3^(n-1).

The arithmetic sequence is defined by the formula an = a + (n-1)d, where a is the first term, d is the common difference, and n is the term number. In this case, the first term a is 100 and the common difference d is -2.5. To find the 6th term, we substitute n=6 into the formula: a6 = 100 + (6-1)(-2.5) = 100 + 5(-2.5) = 100 - 12.5 = 87.5. Therefore, the 6th term of the arithmetic sequence is 87.5.

The arithmetic sequence starts with 4 and increases by 20 each time. To find the sum of the first five terms, we can use the formula for the sum of an arithmetic series: S = (n/2)(2a + (n-1)d), where S is the sum, a is the first term, n is the number of terms, and d is the common difference. Plugging in the values, we get S = (5/2)(2(4) + (5-1)(20)) = 220.

The given sequence is a geometric progression with a common ratio of 1/5. To find the sum to infinity of a geometric progression, we use the formula S = a / (1 - r), where S is the sum to infinity, a is the first term, and r is the common ratio. Plugging in the values, we get S = 250 / (1 - 1/5) = 250 / (4/5) = 312.5. Therefore, the sum to infinity of the sequence is 312.5.

In a geometric sequence, each term is found by multiplying the previous term by a constant ratio. To find the common ratio, we can divide the 4th term (6) by the 1st term (384). This gives us a ratio of 1/64. To find the 6th term, we can multiply the 4th term by the square of the common ratio. Therefore, the 6th term is 6 * (1/64)^2 = 6 * 1/4096 = 1/682.67, which is approximately equal to 0.375.

In an arithmetic sequence with a first term of 4 and a common difference of 2.5, each term can be found by adding 2.5 to the previous term. To find the term number that exceeds 100, we can set up the equation 4 + (n-1) * 2.5 > 100, where n represents the term number. Solving this equation, we find that n = 40, indicating that the first term to exceed 100 is the 40th term.

The geometric sequence is given by the formula an = a * r^(n-1), where a is the first term and r is the common ratio. In this case, a = 4 and r = 2.5. Plugging these values into the formula, we get an = 4 * (2.5)^(n-1). We need to find the term number when an exceeds 100. Setting up the equation 4 * (2.5)^(n-1) > 100 and solving for n, we find that n = 5. Therefore, the first term in this sequence to exceed 100 is term number 5.

Based on the information given, Sarah started with 1000 songs in the first week. In the next week, she added 100 new songs, bringing the total to 1100. In each succeeding week, she continued to add 100 new songs, resulting in a total of 1200 songs in the second week and 1300 songs in the third week. Therefore, the missing values in the table are 1100, 1200, and 1300.

Sarah stored 1000 songs in the first week and added 100 songs each week thereafter. So, by the end of week 6, she would have added 100*(6-1) = 500 songs in total. Adding this to the initial 1000 songs, she will have a total of 1500 songs stored on her MP3 player.

This sequence is an arithmetic sequence because each week Sarah adds the same number of songs (100) to her MP3 player. The difference between each term in the sequence is constant, indicating an arithmetic progression.

Sarah stored 1000 songs in the first week and added 100 new songs each week thereafter. This means that in week 2, she will have 1000 + 100 = 1100 songs. In week 3, she will have 1100 + 100 = 1200 songs. This pattern continues until week 20. Therefore, by the start of week 20, she will have 1000 + (100 * 19) = 2800 songs stored on her MP3 player.

The expression for the number of songs stored on Sarah's MP3 player at the end of the nth week is 900+100n. This can be deduced from the information given in the question that Sarah stored 1000 songs in the first week and added 100 new songs in each succeeding week. Since there are n weeks, the number of songs stored at the end of the nth week can be calculated by multiplying the number of weeks (n) by 100 and adding it to the initial 900 songs that were stored in the first week.

Each week, Sarah adds 100 new songs to her MP3 player. To find out how many weeks it will take for her to reach 5000 songs, we can divide 5000 by 100. The result is 50. However, since she already had 1000 songs stored in the first week, we need to subtract that from the total. So, it will take her 50 weeks to add 5000 songs in total, but since she already had 1000 songs in the first week, it will be 41 weeks from the start to reach 5000 songs.

The company predicts that its sales will decrease geometrically by 25% each year. To find the missing values in the table, we need to decrease the previous year's sales by 25%.

Starting with 10,000 copies sold in the first year, we decrease it by 25% to get 7,500 copies sold in the second year.

For the third year, we decrease 7,500 copies by 25% to get 5,625 copies sold.

Therefore, the missing values in the table are 7,500 and 5,625.

The common ratio in a geometric sequence is the factor by which each term is multiplied to obtain the next term. In this case, the sales are predicted to decrease by 25% each year. Since a decrease of 25% is equivalent to multiplying by 0.75, the common ratio is 0.75.

The number of books sold in the 5th year can be determined by multiplying the initial number of books sold (10,000) by the decreasing factor of 0.75 (100% - 25%). So, the number of books sold in the 5th year would be 10,000 * 0.75 * 0.75 * 0.75 * 0.75 * 0.75 = 3164.

Since the company predicts that its sales will decrease geometrically by 25% each year, we can calculate the sales for each year by multiplying the previous year's sales by 0.75 (100% - 25%).
Starting with 10,000 copies sold in the first year, we can calculate the sales for each subsequent year as follows:
Year 1: 10,000 * 0.75 = 7,500
Year 2: 7,500 * 0.75 = 5,625
Year 3: 5,625 * 0.75 = 4,219
Year 4: 4,219 * 0.75 = 3,164
Year 5: 3,164 * 0.75 = 2,373
Year 6: 2,373 * 0.75 = 1,780
Year 7: 1,780 * 0.75 = 1,335
Year 8: 1,335 * 0.75 = 1,001.25
In year 8, the sales drop below 1,000 copies. Therefore, the correct answer is 10.

The total number of books sold in the first 10 years can be calculated by summing up the number of books sold each year. In the first year, 10,000 books are sold. In the second year, the number of books sold decreases by 25%, so 7,500 books are sold. This pattern continues for the next 8 years, resulting in a total of 37,747 books sold in the first 10 years.

The total number of books that can ever be sold is 40,000. This can be calculated by using the formula for the sum of an infinite geometric series, where the first term is 10,000 and the common ratio is 0.75 (since the sales decrease by 25% each year). The formula is given by a / (1 - r), where a is the first term and r is the common ratio. Plugging in the values, we get 10,000 / (1 - 0.75) = 10,000 / 0.25 = 40,000.

The expression B represents the number of books sold in the nth year of publication. This is because the expression B is a geometric sequence that decreases by 25% each year. The initial number of books sold is 10,000, and each subsequent year the number of books sold is found by multiplying the previous year's sales by 0.75. Therefore, B represents the correct expression for the number of books sold in the nth year.

The sum of the first six terms of a geometric sequence can be found using the formula: S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, the first term is 5 and the common ratio is 4. Plugging these values into the formula, we get S = 5 * (1 - 4^6) / (1 - 4), which simplifies to S = 5 * (1 - 4096) / (1 - 4) = 5 * (-4095) / (-3) = 6825. Therefore, the sum of the first six terms is 6825.