JEE Main 2020 (Online) 3rd September Morning ShiftQuestion 1
If the number of integral terms in the expansion of $\left(3^{\frac12}+5^{\frac18}\right)^n$ is exactly $33$, then the least value of $n$ is :
Binomial Theorem · Rational and Integral Terms
Solution
Concept. A term is an integer when both exponents are whole numbers; count the valid $r$. Formulas used. (1) $T_{r+1}={}^{n}C_r\,a^{n-r}b^r$Step 1 - General term. $$T_{r+1}={}^{n}C_r\,3^{\frac{n-r}{2}}\,5^{\frac r8}.$$ Step 2 - Conditions. We need $8\mid r$ and $2\mid(n-r)$. If $8\mid r$ then $r$ is even, so $n$ must be even. (For odd $n$ there are no integral terms.) Step 3 - Count. For even $n$ the integral terms are $r=0,8,16,\dots$, so there are $\left\lfloor\frac n8\right\rfloor+1$ of them. This equals $33$ when $\left\lfloor\frac n8\right\rfloor=32$, i.e. $256\le n\le263$. Step 4 - Least $n$. $n=256$ is even, so it works. Answer: (C)