JEE MAIN - Mathematics (2025 - 8th April Evening Shift - No. 20)

The number of integral terms in the expansion of $ \left( {5^\frac{1}{2}} + 7^\frac{1}{8} \right)^{1016} $ is:
127
128
130
129

Explanation

$$\begin{aligned} &\begin{aligned} & \mathrm{T}_{\mathrm{r}}={ }^{1016} \mathrm{C}_{\mathrm{r}}(5)^{\frac{1016-\mathrm{r}}{2}} 7^{\frac{\mathrm{r}}{8}} \\ & \Rightarrow \mathrm{r}=0,8,16,24, \ldots ., 1016 \\ & 1016=0+(\mathrm{n}-1) 8 \\ & \Rightarrow \mathrm{n}-1=\frac{1016}{8}=127 \end{aligned}\\ &\text { So, } \mathrm{n}=128 \text {. } \end{aligned}$$

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