QJEEM23Mad95c: JEE Main 2023 Mathematics - Class 11 JEEM23Mad95c Solution: The number of integral terms in the expansion of is equal to…

Question

Verified Solution • 2025–26 Session moderate

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

Accepted Answer

\begin{aligned}&amp; \text { General term of the expansion }\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680} \\\\&amp; \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}

The term will be integral if

r

is a multiple of 4 .

\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}

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