Relationship between two GPs

8 A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(R\) and sum to infinity \(2 S\).

1. Show that \(r = 2 R - 1\).
   It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
2. Express \(S\) in terms of \(a\).

10

1. In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is \(a\).
   1. Show that the common difference of the progression is \(\frac { 1 } { 3 } a\).
   2. Given that the tenth term is 36 more than the fourth term, find the value of \(a\).
2. The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12 , find the value of the fifth term.

6

1. The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200 . Find the seventh term.
2. A geometric progression has first term 1 and common ratio \(r\). A second geometric progression has first term 4 and common ratio \(\frac { 1 } { 4 } r\). The two progressions have the same sum to infinity, \(S\). Find the values of \(r\) and \(S\).

9

1. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 . Find the common difference and the first term.
2. A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity 6. A second geometric progression has first term \(2 a\), common ratio \(r ^ { 2 }\) and sum to infinity 7 . Find the values of \(a\) and \(r\).

3

1. A geometric progression has first term \(3 a\) and common ratio \(r\). A second geometric progression has first term \(a\) and common ratio \(- 2 r\). The two progressions have the same sum to infinity. Find the value of \(r\).
2. The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first \(n\) terms. Find the value of \(n\).

11 A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25 .

1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\).
2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2 ^ { n - 2 } : 3 ^ { n - 2 }\).

11 In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).

Three geometric progressions, \(P\), \(Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2, 1, \frac{1}{2}, \frac{1}{4}, \ldots\) Progression \(Q\) is \(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)

1. Find the sum to infinity of progression \(R\). [3]
2. Given that the first term of \(R\) is 4, find the sum of the first three terms of \(R\). [3]