WAEC 2013 · Paper 2 · Q1

Sequences, series & binomial expansion

(a)
If the coefficients of $x^2$ and $x^3$ in the expansion of $(p + qx)^7$ are equal, express $q$ in terms of $p$ .
(b)
A man makes a weekly contribution into a fund. In the first week, he paid ₦180.00, the second week ₦260.00, the third week ₦340.00 and so on. How much would he have contributed in 16 weeks?

Worked solution (try it first)

(a)

The $x^2$ term of $(p + qx)^7$ is $\binom72p^5(qx)^2 = 21p^5q^2x^2$ .
The $x^3$ term is $\binom73p^4(qx)^3 = 35p^4q^3x^3$ .
Set the coefficients equal: $21p^5q^2 = 35p^4q^3$ .
Divide both sides by $7p^4q^2$ : $3p = 5q$ , so $q = \frac35p$ .

(b)

The payments form an A.P. with $a = 180$ and $d = 80$ .
The total in 16 weeks is $S_{16}$ .
$S_{16} = \dfrac{16}{2}[2(180) + 15(80)]$
$= 8(360 + 1200)$
$= 8 \times 1560$ .
So he contributes ₦12 480.

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