Question

write down the first three terms of a binomial expansion (1+ax)^n in ascending powers of x. if the co efficient of x and x^2 are 2 and 3.5 respectively. find the values of a and n

Answer 1

Answer:

$(1 + ax) {}^{n} = \binom{n}{0} {(ax)}^{0} + \binom{n}{1} {(ax)}^{1} + \binom{n}{2} {(ax)}^{2} + ... \\ \\ = 1 + nax + \frac{n(n - 1)}{2} . {a}^{2} {x}^{2} ....$

Compare coefficients

$an = 2 \: \: \implies \: a = \frac{2}{n}$

And

$\frac{n(n - 1)}{2} \times {a}^{2} = \frac{3}{2} \\ \\ \frac{n(n - 1)}{2} \times \frac{4}{ {n}^{2} } = \frac{3}{2} \\ \\ 4(n - 1) = 3n \\ n = 4$

And a = 2/n = 2/4 = 1/2