The number of integral terms in the expansion of (5^(1)/(2)+7^(1)/(8))^1016 is:

Question

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

Accepted Answer

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

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