Question

In the expansion of ${(5^{\frac{1}{2}}+7^{\frac{1}{8}})}^{1024}$, the number of integral terms is .

• A. 128
• B. 129
• C. 130
• D. 131

Answer 1

The correct option is B 129

Here, $n=1024=2^{10}$

Now, $T_{r+1}={{}^{n}C}_r{\left(5^{\frac{1}{2}}\right)}^{1024-r}{\left(7^{\frac{1}{8}}\right)}^r$

For $T_{r+1}$ to be an integer,
$1024-r$ must be a multiple of 2, or 0.
$r$ must be a multiple of 8, or 0.

When r is a multiple of 8, say $r=8k$,
$1024-r=1024-8k$, which is divisible by 2.

Thus, the number of multiples of 8 between 0 and 1024, including 0 is the number of integers in the expansion.

This number is given by $1024÷8+1=128+1=129$