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JEE Main 20258 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
8 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2025 · 8 April · Evening Shift · Mathematics · Binomial Theorem

The number of integral terms in the expansion of (5^ (1)/(2) +7^...

The number of integral terms in the expansion of (512+718)1016\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)^{1016} is

Options

  1. A127
  2. B130
  3. C129
  4. D128

Correct answer

D. 128

Step-by-step solution

Tr=1016Cr(5)1016−r27r8⇒r=0,8,16,24,….,10161016=0+(n−1)8⇒n−1=10168=127\begin{aligned} & \mathrm{T}_{\mathrm{r}}={ }^{1016} \mathrm{C}_{\mathrm{r}}(5)^{\frac{1016-\mathrm{r}}{2}} 7^{\frac{\mathrm{r}}{8}} \\ & \Rightarrow \mathrm{r}=0,8,16,24, \ldots ., 1016 \\ & 1016=0+(\mathrm{n}-1) 8 \\ & \Rightarrow \mathrm{n}-1=\frac{1016}{8}=127 \end{aligned} So, n=128\mathrm{n}=128.
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