The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Binomial Theorem

The number of integral terms in the expansion of

(3^{\frac{1}{2}} + 5^{\frac{1}{4}})^{680}

is equal to

Answer: 171

The general term in the expansion is

T_{r} = (r 680) (3^{1/2})^{680 - r} (5^{1/4})^{r} = (r 680) 3^{(680 - r) /2} 5^{r /4}

. For the term to be integral, both exponents must be integers:

(680 - r) /2

is an integer for all

r

, and

r /4

is an integer when

r

is a multiple of 4. Thus

r \in {0, 4, 8, 12, \dots, 680}

. The number of such values is

\frac{680 - 0}{4} + 1 = 171

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Integer answer