20. A sequence is given as \( 6(3)^{n-2} \).

(a) Write down the first 6 terms of the sequence. (3 marks)

For \( n = 1, 2, 3, 4, 5, 6 \):
\(
6(3)^{-1} = 2, \quad 6(3)^0 = 6, \quad 6(3)^1 = 18, \quad 6(3)^2 = 54, \quad 6(3)^3 = 162, \quad 6(3)^4 = 486.
\)

First 6 terms: \( 2, 6, 18, 54, 162, 486 \).

(b) Find the sum of the first 10 terms of the sequence. (3 marks)

Given \( a = 2 \) and \( r = 3 \):
\(
S_n = \frac{a(r^n - 1)}{r - 1}.
\)

For \( n = 10 \):
\(
S_{10} = \frac{2(3^{10} - 1)}{3 - 1} = \frac{2(59048)}{2} = 59048.
\)

(c) Find the values of \( n \) for which the sum of the first \( n \) terms is at most 728. (4 marks)

We solve:
\(
\frac{2(3^n - 1)}{3 - 1} \leq 728 \quad \Rightarrow \quad 3^n - 1 \leq 728 \quad \Rightarrow \quad 3^n \leq 729.
\)

Since \( 3^6 = 729 \):
\(
n \leq 6.
\)

Therefore, \( n \leq 6 \).

KCSE 2002, PP2, Q14
Each month, for 40 months, Amina deposited some money in a saving scheme. In the 5th month she deposited sh 500. Thereafter she increased her deposits by sh 50 every month. Calculate the:
(a) last amount deposited by Amina (2 marks)
(b) total amount Amina had saved in the 40 months. (2 marks)

KCSE 2004, PP2, Q3
Find the number of terms of the series
2+6+10+14+18+…2 + 6 + 10 + 14 + 18 + \(\dots\) that will give a sum of 800. (2 marks)

KCSE 2007, PP2, Q10
A carpenter wishes to make a ladder with 15 cross-pieces. The cross-pieces are to diminish uniformly in lengths from 67 cm at the bottom to 32 cm at the top. Calculate the length, in cm, of the seventh cross-piece from the bottom. (3 marks)

KCSE 2013, PP2, Q1
The sum of nn terms of sequence:
3,9,15,21,…3, 9, 15, 21, \(\dots\)  is 7,500. Determine the value of nn. (3 marks)

KCSE 2017, PP2, Q23
(a) The 5th term of an AP is 82 and the 12th term is 103.
Find:
(i) the first term and the common difference; (3 marks)
(ii) the sum of the first 21 terms. (2 marks)

(b) A staircase was built such that each subsequent stair has a uniform difference in height. The height of the 6th stair from the horizontal floor was 85 cm and the height of the 10th stair was 145 cm. Calculate the height of the 1st stair and the uniform difference in height of the stairs. (3 marks)

KCSE 2018, PP2, Q17
The 5th and 10th terms of an arithmetic progression are 18 and –2 respectively.
(a) Find the common difference and the first term.
(4 marks)

 (b) Determine the least number of terms which must be added together so that the sum of the progression is negative. Hence find the sum. (6 marks)

KCSE 1997, PP1, Q4
In a geometric progression, the first term is a and the common ratio is r. The sum of the first two terms is 12 and the third term is 16.
(a) Determine the ratio  \(\frac{ar^2}{(a+ar)}\)  (1 mark)
(b) If the first term is larger than the second term, find the value of r. (2 marks)

KCSE 2003, PP2, Q15
A colony of insects was found to have 250 insects at the beginning. Thereafter the number of insects doubled every 2 days. Find how many insects there were after 16 days. (3 marks)

KCSE 2005, PP2, Q5
The first three consecutive terms of a geometrical progression are 3, \(x\), and \(5\frac{1}{3}\). Find the value of x. (2 marks)

KCSE 2005, PP2, Q19
Abdi and Amoit were employed at the beginning of the same year. Their annual salaries, in shillings, progressed as follows:
Abdi: 60 000, 64 800, 69 600, ...
Amoit: 60 000, 64 800, 69 984, ...

(a) Calculate Abdi’s annual salary increment and hence write down an expression for his annual salary in his nth year of employment. (2 marks)

 (b) Calculate Amoit’s annual percentage rate of salary increment and hence write down an expression for her annual salary in her nth year of employment. (2 marks)

 (c) Calculate the difference in the annual salaries for Abdi and Amoit in their 7th year of employment. (4 marks)

KCSE 2011, PP2, Q18 Tested again in 2016, PP2, Q24
The first, fifth and seventh terms of an arithmetic progression (A.P) correspond to the first three consecutive terms of a decreasing geometric progression (G.P). The first term of each progression is 64, the common difference of the A.P is d and the common ratio of the G.P is r.
(a)
(i) Write two equations involving d and r. (2 marks)
(ii) Find the values of d and r. (4 marks)
(b) Find the sum of the first 10 terms of:
(i) The arithmetic progression (A.P) (2 marks)
(ii) The geometric progression (G.P) (2 marks)