The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to ___________.
Answer (integer)
171
Solution
$$
\begin{aligned}
& \text { General term of the expansion }\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680} \\\\
& \qquad={ }^{680} C_r\left(3^{1 / 2}\right)^{680-r}\left(5^{1 / 4}\right)^r={ }^{680} C_r \times 3^{\frac{680-r}{2}} \times 5^{\frac{r}{4}}
\end{aligned}
$$
<br/><br/>The term will be integral if $r$ is a multiple of 4 .
<br/><br/>$$
\begin{gathered}
\therefore r=0,4,8,12, \ldots, 680(\text { which is an } \mathrm{AP}) \\\\
680=0+(n-1) 4 \\\\
n=\frac{680}{4}+1=171
\end{gathered}
$$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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