JEE Main 2025 Apr 8 Shift 2, Mathematics Q16: Introduction to Binomial Expansion

JEE Main2025Apr 8, Shift 2Mathematics

The number of integral terms in the expansion of $(5^{1/2} + 7^{1/8})^{1016}$ is

Solution

The general term is $T_{r + 1} = (r 1016) 5^{(1016 - r) /2} \cdot 7^{r /8}$ .

For this term to be an integer, both exponents $(1016 - r) /2$ and $r /8$ must be non-negative integers.

$(1016 - r) /2$ is an integer iff $r$ is even; $r /8$ is an integer iff $8 ∣ r$ . Combined, $r$ must be a multiple of 8.

Valid values of $r$ : $0, 8, 16, \dots, 1016$ , forming an arithmetic progression with common difference 8.

Number of terms: $\frac{1016 - 0}{8} + 1 = 127 + 1 = 128$ , matching option (4).

Concept behind this question

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