1.04i Geometric sequences: nth term and finite series sum

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

1. Find the common difference of the progression. [2]

The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

1. Find the common ratio of the geometric progression and the value of \(n\). [5]

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic progression.

1. Write down an expression for his total weight loss after \(x\) weeks. [1]
2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\). [2]

Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.

1. Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

1. an arithmetic progression, [2]
2. a geometric progression. [2]

The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.

1. Find the value of \(x\) and the value of \(y\). [4]
2. Find the fourth term of each progression. [3]

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by 2% of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.

1. Find the amount of salt obtained in the 12th week after the change. [3]
2. Find the total amount of salt obtained in the first 12 weeks after the change. [2]

The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]

Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum_{n=1}^N 2^{-n}z^n.$$ [2] Use de Moivre's theorem to deduce that $$\sum_{n=1}^{10} 2^{-n}\sin\left(\frac{1}{10}n\pi\right) = \frac{1025\sin\left(\frac{1}{10}\pi\right)}{2560 - 2048\cos\left(\frac{1}{10}\pi\right)}.$$ [6]

In a geometric sequence \(u_1, u_2, u_3, \ldots\)

• the common ratio is \(r\)
• \(u_2 + u_3 = 6\)
• \(u_4 = 8\)

1. Show that \(r\) satisfies $$3r^2 - 4r - 4 = 0$$ [3]

Given that the geometric sequence has a sum to infinity,

1. find \(u_1\) [3]
2. find \(S_∞\) [2]

The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find

1. the common ratio, [2]
2. the first term, [2]
3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]

1. A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac{a(1-r^n)}{1-r}.$$ [4]

Mr King will be paid a salary of £35 000 in the year 2005. Mr King's contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.

1. Find, to the nearest £100, Mr King's salary in the year 2008. [2]

Mr King will receive a salary each year from 2005 until he retires at the end of 2024.

1. Find, to the nearest £1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. [4]

The first term of a geometric series is 120. The sum to infinity of the series is 480.

1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
2. Find, to 2 decimal places, the difference between the 5th and 6th terms. [2]
3. Calculate the sum of the first 7 terms. [2]

The sum of the first \(n\) terms of the series is greater than 300.

1. Calculate the smallest possible value of \(n\). [4]

A trading company made a profit of £50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r, r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of £50 000r will be made.

1. Write down an expression for the predicted profit in Year \(n\). [1]

The model predicts that in Year \(n\), the profit made will exceed £200 000.

1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]

Using the model with \(r = 1.09\),

1. find the year in which the profit made will first exceed £200 000, [2]
2. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest £10 000. [3]

The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find

1. the common ratio, [2]
2. the first term, [2]
3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]

The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find

1. the common ratio, [2]
2. the first term, [2]
3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]

The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find

1. the value of the common ratio, [3]
2. the first term, [2]
3. the sum to infinity. [2]

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}.$$ [4]

The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find

1. the value of \(r\) and the value of \(a\), [5]
2. the sum to infinity of the series. [3]

A geometric series has first term 1200. Its sum to infinity is 960.

1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
3. Write down an expression for the sum of the first \(n\) terms of the series. [2]

Given that \(n\) is odd,

1. prove that the sum of the first \(n\) terms of the series is $$960(1 + 0.25^n).$$ [2]

The first three terms of an arithmetic series are \((12 - p)\), \(2p\) and \((4p - 5)\) respectively, where \(p\) is a constant.

1. Find the value of \(p\). [2]
2. Show that the sixth term of the series is 50. [3]
3. Find the sum of the first 15 terms of the series. [2]
4. Find how many terms of the series have a value of less than 400. [3]

A geometric series has second term \(375\) and fifth term \(81\).

1. 1. Show that the common ratio of the series is \(0.6\). [3]
   2. Find the first term of the series. [2]
2. Find the sum to infinity of the series. [2]
3. The \(n\)th term of the series is \(u_n\). Find the value of \(\sum_{n=6}^{\infty} u_n\). [4]

A geometric series has first term \(1200\). Its sum to infinity is \(960\).

1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
3. Write down an expression for the sum of the first \(n\) terms of the series. [2]

Given that \(n\) is odd,

1. prove that the sum of the first \(n\) terms of the series is $$960(1 + 0.25^n).$$ [2]