Sum of first n terms

2 Find

1. the sum of the first ten terms of the geometric progression \(81,54,36 , \ldots\),
2. the sum of all the terms in the arithmetic progression \(180,175,170 , \ldots , 25\).

1. Evaluate

$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$

8 \begin{figure}[h] \end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).

1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. (a) Find the area of the fifth sector in the pattern.
   (b) Find the total area of the first ten sectors in the pattern.
   (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.

7

1. The first term of a geometric progression is 7 and the common ratio is - 2 .
   1. Find the ninth term.
   2. Find the sum of the first 15 terms.
2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).

1 A geometric progression has first term 3 and second term - 6 .

1. State the value of the common ratio.
2. Find the value of the eleventh term.
3. Find the sum of the first twenty terms.

2 The \(n\)th term of a geometric sequence is \(u _ { n }\), where $$u _ { n } = 3 \times 4 ^ { n }$$

1. Find the value of \(u _ { 1 }\) and show that \(u _ { 2 } = 48\).
2. Write down the common ratio of the geometric sequence.
3. 1. Show that the sum of the first 12 terms of the geometric sequence is \(4 ^ { k } - 4\), where \(k\) is an integer.
   2. Hence find the value of \(\sum _ { n = 2 } ^ { 12 } u _ { n }\).

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]

A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5. Find the value of \(b\) and find also the sum of the first 15 terms of the progression. [5]

A series is given by $$\sum_{r=1}^k 9^{r-1}$$

1. Write down a formula for the sum of this series. [1]
2. Prove by induction that \(P(n) = 9^n - 8n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1. [4]

The table shows information about three geometric series. The three geometric series have different common ratios.

1. Find \(n_1\). [2]
2. Given that \(r_2\) is an integer less than 10, find the value of \(r_2\) and the value of \(n_2\). [2]
3. Given that \(r_3\) is not an integer, find a possible value for the sum of all the terms in Series 3. [4]