Correct Answer - Option 1 : 38
Given term is \({\left( {{x^2} + \frac{1}{{{x^3}}}} \right)^n}\)
General term,
\(\;{T_{r + 1}} = {}_\;^n{C_r}{\left( {{x^2}} \right)^{n - r}}{\left( {\frac{1}{{{x^3}}}} \right)^r} = {}_\;^n{C_r}\cdot{x^{2n - 5r}}\)
To find coefficient of x, 2n - 5r = 1
∵ 2n - 5r = 1
⇒ 2n = 5r + 1
\(\therefore r = \frac{{2n - 1}}{5}\)
\(\therefore {\rm{\;Coefficient\;of\;}}x = {}_\;^n{C_{\left( {\frac{{2n - 1}}{5}} \right)}} = {}_\;^n{C_{23}}\)
\(\therefore \frac{{2{\rm{n}} - 1}}{5} = 23{\rm{\;\;\;\;or\;\;\;n}} - \left( {\frac{{2{\rm{n}} - 1}}{5}} \right) = 23\)
2n - 1 = 115 or 3n + 1 = 115
2n = 116 or 3n = 114
n = 58 or n = 38
∴ The smallest value of n = 38
