Write down the first three terms of the binomial expansion ((1 + ax)^{n}) in ascending powers of x. If the coefficients of x and x(^{2}) are 2 and (frac{3}{2}) respectively, find the values of a and n.
Explanation
((1 + ax)^{n} = ^{n}C_{n} (1)^{n} + ^{n}C_{n – 1} (1)^{n – 1} (ax) + ^{n}C_{n – 2} (1)^{n – 2} (ax)^{2} + …)
= (1 + nax + (frac{n(n – 1)}{2})(ax)^{2} + … )
Given
(an = 2 …. (1))
(frac{n^{2} – n}{2}) a^{2} = frac{3}{2} … (2))
(frac{a^{2}n^{2} – a^{2} n}{2} = frac{3}{2})
(frac{(an)^{2} – a(an)}{2} = frac{3}{2} … (3))
From (1), (an = 2). (3) becomes
(frac{2^{2} – a(2)}{2} = frac{3}{2})
(frac{4 – 2a}{2} = frac{3}{2})
(2 – 2a = frac{3}{2} implies 2a = frac{1}{2})
(therefore a = frac{1}{4})
Recall (an = 2)
(frac{n}{4} = 2 implies n = 8)