The number of integral terms in the expansion of ( 3 1 / 2 + 5 1 / 4 ) 680 is equal to ___________ . [2023]

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Solution

Q.
The number of integral terms in the expansion of ${(3^{1 / 2} + 5^{1 / 4})}^{680}$ is equal to ___________ . [2023]

Ans.
(171)

The expansion of the general term is

$= {}^{680}{C_{r}} {(3^{1 / 2})}^{680 - r} {(5^{1 / 4})}^{r} = {}^{680}{C_{r}} 3^{\frac{680 - r}{2}} 5^{\frac{r}{4}}$

Possible values of $r$ , where $\frac{r}{4}$ is an integer

$r = 0, 4, 8, 12, \dots, 680$

All these values of $r$ are accepted as well.

$Hence, number of integral terms = 171$

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