WAEC 2008 · Paper 2 · Q13✱✱

Sequences & series (AP, GP)

(a)
The first term of an arithmetic progression (A.P.) is $31$ and the common difference is $9$ . Show that the $n$ th term is $9n + 22$ . Hence, find the 20th term.
(b)
The second and fifth terms of a geometric progression (G.P.) are $1$ and $\frac18$ respectively. Find the: (i) common ratio; (ii) first term; (iii) eighth term.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

The $n$ th term of an A.P. is $a + (n - 1)d = 31 + 9(n - 1)$ .
Expand: $31 + 9n - 9 = 9n + 22$ , as required.
Hence the 20th term is $9(20) + 22 = 202$ .

(b)(i)

The second term is $ar = 1$ and the fifth term is $ar^4 = \frac18$ .
Divide the fifth term by the second: $r^3 = \frac18$ , so the common ratio is $r = \frac12$ .

(ii)

From $ar = 1$ : $a = 1 \div \frac12 = 2$ .
The first term is $2$ .

(iii)

The eighth term is $ar^7 = 2 \times \left(\frac12\right)^7$
$= \frac{2}{128}$
$= \frac{1}{64}$ .

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