The number of integral terms in the expansion of (3^(1)/(2)+5^(1)/(4))^680 is equal to ___…

Question

The number of integral terms in the expansion of ${({3}^{\frac{1}{2}}+{5}^{\frac{1}{4}})}^{680}$ is equal to \_\_\_\_\_\_\_\_\_\_\_.

Answer: 171

Accepted Answer

$\begin{gathered} {({3}^{\frac{1}{2}}+{5}^{\frac{1}{4}})}^{680} \ \, ={}^{680}{C}_{r}{({3}^{1/2})}^{680-r}{({5}^{1/4})}^{r}={}^{680}{C}_{r}	imes {3}^{\frac{680-r}{2}}	imes {5}^{\frac{r}{4}} \end{gathered}$

The term will be integral if $r$ is a multiple of 4 .

$\begin{gathered} 	herefore r=0,4,8,12,\ldots ,680(AP) \ 680=0+(n-1)4 \ n=\frac{680}{4}+1=171 \end{gathered}$

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