Find N for S_∞ - S_N condition

3 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

1. The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first \(m\) terms is zero. Find the value of \(m\).
2. A geometric progression, in which all the terms are positive, has common ratio \(r\). The sum of the first \(n\) terms is less than \(90 \%\) of the sum to infinity. Show that \(r ^ { n } > 0.1\).

1. The first term of a geometric series is 6 and the common ratio is 0.92

For this series, find

1. 1. the \(25 ^ { \text {th } }\) term, giving your answer to 2 significant figures,
   2. the sum to infinity. The sum to \(n\) terms of this series is greater than 72
2. Calculate the smallest possible value of \(n\).
   VJYV SIHI NITIIIUMION, OC
   VILV SIHI NI JAHM ION OO
   VILV SIHI NI JIIIM ION OO

1. A geometric sequence has first term \(a\) and common ratio \(r\), where \(r > 0\)

Given that

• the 3rd term is 20
• the 5th term is 12.8
  1. show that \(r = 0.8\)
  2. Hence find the value of \(a\).

Given that the sum of the first \(n\) terms of this sequence is greater than 156

find the smallest possible value of \(n\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)

2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity of the series is \(S _ { \infty }\)

1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
3. Find the smallest value of \(N\), for which \(S _ { \infty } - S _ { N } < 0.5\) 2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity
   of the series is \(S _ { \infty }\)

6. A geometric series has first term 5 and common ratio \(\frac { 4 } { 5 }\). Calculate

1. the 20th term of the series, to 3 decimal places,
2. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
3. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
4. find the smallest possible value of \(k\).

2. A geometric series has first term \(a\), where \(a \neq 0\), and common ratio \(r\). The sum to infinity of this series is 6 times the first term of the series.

1. Show that \(r = \frac { 5 } { 6 }\) Given that the fourth term of this series is 62.5
2. find the value of \(a\),
3. find the difference between the sum to infinity and the sum of the first 30 terms, giving your answer to 3 significant figures.

6. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\) The sum to infinity of the series is \(S _ { \infty }\)

1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
3. Find the smallest value of \(N\), for which $$S _ { \infty } - S _ { N } < 0.5$$

5 In a geometric progression, the first term is 5 and the second term is 4.8 .

1. Show that the sum to infinity is 125 .
2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).

8 The first term of a geometric progression is 10 and the common ratio is 0.8.

1. Find the fourth term.
2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002 ,$$ and use logarithms to find the smallest possible value of \(N\).

7 In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]

In this question you must show detailed reasoning. A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]

The first term of a geometric progression is \(10\) and the common ratio is \(0.8\).

1. Find the fourth term. [2]
2. Find the sum of the first \(20\) terms, giving your answer correct to \(3\) significant figures. [2]
3. The sum of the first \(N\) terms is denoted by \(S_N\), and the sum to infinity is denoted by \(S_\infty\). Show that the inequality \(S_\infty - S_N < 0.01\) can be written as $$0.8^N < 0.0002,$$ and use logarithms to find the smallest possible value of \(N\). [6]

The first term of a geometric progression is 12 and the second term is 9.

1. Find the fifth term. [3]

The sum of the first \(N\) terms is denoted by \(S_N\) and the sum to infinity is denoted by \(S_\infty\). It is given that the difference between \(S_\infty\) and \(S_N\) is at most 0.0096.

1. Show that \(\left(\frac{3}{4}\right)^N \leqslant 0.0002\). [3]
2. Use logarithms to find the smallest possible value of \(N\). [2]