The smallest natural number n, such that the coefficient of x in the expansion of [( x^2 +…

Question

The smallest natural number n, such that the coefficient of x in the expansion of $${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$$ is nC23, is :

Accepted Answer

General term

$${T_{r + 1}} = {}^n{C_r}{x^{2n - 2r}}.{x^{ - 3r}}$$

$$ \therefore $$ $$2n - 5r = 1 \Rightarrow 2n = 5r + 1$$

$$ \therefore $$ $$r = {{2n - 1} \over 5}$$

$$ \Rightarrow $$ Coefficient of x = $${}^n{C_{\left( {{{2n - 1} \over 5}} \right)}} = {}^n{C_{23}}$$

$$ \Rightarrow $$ $${{2n - 1} \over 5} = 23\,\,or\,\,n - \left( {{{2n - 1} \over 5}} \right) = 23$$

$$ \Rightarrow $$ 2n - 1 = 115 $$ \Rightarrow $$ n = 58

and n = 38

$$ \therefore $$ smallest n = 38

Solutions · 1