Find first term from conditions

6

1. A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term.
2. An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms.

8

1. A geometric progression has a second term of 12 and a sum to infinity of 54 . Find the possible values of the first term of the progression.
2. The \(n\)th term of a progression is \(p + q n\), where \(p\) and \(q\) are constants, and \(S _ { n }\) is the sum of the first \(n\) terms.
   1. Find an expression, in terms of \(p , q\) and \(n\), for \(S _ { n }\).
   2. Given that \(S _ { 4 } = 40\) and \(S _ { 6 } = 72\), find the values of \(p\) and \(q\).

9

1. A geometric progression has first term 100 and sum to infinity 2000. Find the second term.
2. An arithmetic progression has third term 90 and fifth term 80 .
   1. Find the first term and the common difference.
   2. Find the value of \(m\) given that the sum of the first \(m\) terms is equal to the sum of the first ( \(m + 1\) ) terms.
   3. Find the value of \(n\) given that the sum of the first \(n\) terms is zero.

5 The first term of a geometric series is 8. The sum to infinity of the series is 10 .
Find the common ratio.

5 In a geometric progression, the sum to infinity is four times the first term.

1. Show that the common ratio is \(\frac { 3 } { 4 }\).
2. Given that the third term is 9 , find the first term.
3. Find the sum of the first twenty terms.

2 A geometric series has first term 3. The sum to infinity of the series is 8 .
Find the common ratio.

5 The sum to infinity of a geometric series is four times the first term of the series.

1. Show that the common ratio, \(r\), of the geometric series is \(\frac { 3 } { 4 }\).
2. The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
3. The \(n\)th term of the geometric series is \(u _ { n }\) and the ( \(2 n\) )th term of the series is \(u _ { 2 n }\).
   1. Write \(u _ { n }\) and \(u _ { 2 n }\) in terms of \(n\).
   2. Hence show that \(\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)\).
   3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.

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 term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]

A geometric series has common ratio \(\frac{1}{3}\). Given that the sum of the first four terms of the series is 200,

1. find the first term of the series, [3]
2. find the sum to infinity of the series. [2]

The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]

The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18

1. Find the first term and common ratio of the series. [5 marks]
2. 1. Show that the \(n\)th term of the series, \(u_n\), can be written as $$u_n = \frac{3^n}{2^{2n-5}}$$ [4 marks]
   2. Hence show that $$\log_3 u_n = n(1 - 2\log_3 2) + 5\log_3 2$$ [3 marks]

The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q\sqrt{r}\) where \(p\), \(q\) and \(r\) are integers. [6]