(a) If the coefficient of \(x^{2}\) and \(x^{3}\) in the expansion of \((p + qx)^{7}\)
(a) If the coefficient 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 N180.00, second week N260.00, third week N340.00 and so on. How much would he have contributed in 16 weeks?
Explanation
(a) \((p + qx)^{7} = p^{7} + 7p^{6} qx + 21p^{5} (qx)^{2} + 35p^{4} (qx)^{3} + 35p^{3} (qx)^{4} + 21p^{2} (qx)^{5} + 7p(qx)^{6} + (qx)^{7}\)
The coefficient of \(x^{2}\) = \(21p^{5} q^{2}\)
The cofficient of \(x^{3}\) = \(35p^{4} q^{3}\)
\(\therefore 21p^{5} q^{2} = 35p^{4} q^{3}\)
\(\implies 21p = 35q\)
\(q = \frac{21}{35} p = \frac{3}{5} p\).
(b) N180 + N260 + N340 + ...
This sequence is a linear sequence with first term = a = 180, d = 80.
\(S_{16} = ?\)
\(S_{n} = \frac{n}{2} (2a + (n - 1) d)\)
\(S_{16} = \frac{16}{2} (2(180) + (16 - 1) 80\)
= \(8(360 + 1200)\)
= \(8(1560)\)
= \(N12,480\)
He would have contributed N12,480 by the 16th week.