The number of integral terms in the expansion of is [2025]

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Solution

Q.
The number of integral terms in the expansion of ${(5^{\frac{1}{2}} + 7^{\frac{1}{8}})}^{1016}$ is [2025]

1 129

2 127

3 130

4 128

Ans.
(4)

$T_{r + 1} = {}^{1016}C_{r} 5^{\frac{1016 - r}{2}} 7^{r / 8}$ , represents the ${(r + 1)}^{th}$ term of ${(5^{1 / 2} + 7^{1 / 8})}^{1016}$ .

As $T_{r + 1}$ represent an integral term when r is a multiple of 8

i.e., r = 0, 8, 16, 24, ....., 1016

Now, 1016 = 0 + (n – 1)8 [As this form an A.P.]

$\Rightarrow n - 1 = 127 \Rightarrow n = 128$

So, 128 terms are integral terms.

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