Animated Solution for Mathematics - Binomial Theorem: The number of integral terms in the expansion of $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1016}$ is

The expression $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1016}$ represents a binomial expansion with $1017$ individual terms. Brute force is not the path to the solution; instead, we must identify the conditions under which the terms become integers.

Every term in a binomial expansion follows the General Term formula:

Substituting our specific values where $n=1016$, $a=5^{\frac{1}{2}}$, and $b=7^{\frac{1}{8}}$, we obtain:

The binomial coefficient $\left(\genfrac{}{}{0ex}{}{1016}{r}\right)$ is always an integer. Therefore, for $T_{r+1}$ to be an integer, the exponents of $5$ and $7$ must be non-negative integers.

We must satisfy two primary conditions to eliminate fractional powers:

1. $\frac{1016-r}{2}\in \mathbb{Z}$ 2. $\frac{r}{8}\in \mathbb{Z}$

For the second condition, $r$ must be a multiple of $8$. This implies $r\in \{0,8,16,\dots ,1016\}$.

For the first condition, $1016-r$ must be even. Since $1016$ is even, $r$ must also be even. Because every multiple of $8$ is inherently even, satisfying the second condition automatically satisfies the first.

We need to count the number of values of $r$ in the range $0\le r\le 1016$ that are multiples of $8$. This forms an Arithmetic Progression:

Using the formula for the $n$-th term of an A.P., $l=a+(n-1)d$, where $a=0$, $d=8$, and $l=1016$:

Solving for $n$:

There are exactly $128$ integral terms in the expansion.

The "LCM Rule" confirms this: the index $r$ must be a multiple of the Least Common Multiple of the denominators of the fractional powers. Since $\text{lcm}(2,8)=8$, $r$ must be a multiple of $8$, leading directly to our result.