WAEC 2019 · Paper 2 · Q3

Sequences, series & binomial expansion

(a)
How many terms of the series $-3 - 1 + 1 + \ldots$ add up to 165?

Worked solution (try it first)

The first term is $a = -3$ and the common difference is $d = -1 - (-3) = 2$ .
$S_n = \dfrac n2[2(-3) + (n - 1)(2)]$
$= \dfrac n2(2n - 8)$
$= n(n - 4)$ .
Set $n(n - 4) = 165$ : $n^2 - 4n - 165 = 0$ .
Factorise: $(n - 15)(n + 11) = 0$ , so $n = 15$ or $n = -11$ .
A number of terms can't be negative, so 15 terms add up to 165.

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