(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.