The sum of all rational terms in the expansion of ( 2 1 5 + 5 1 3 ) 15 is equal to [2024]

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Solution

Q.
The sum of all rational terms in the expansion of ${(2^{\frac{1}{5}} + 5^{\frac{1}{3}})}^{15}$ is equal to [2024]

1 3133

2 633

3 931

4 6131

Ans.
(1)

We have, ${(2^{1 / 5} + 5^{1 / 3})}^{15}$

$∴$ $T_{r + 1} = {}^{15}{C_{r}} {(2^{1 / 5})}^{15 - r} {(5^{1 / 3})}^{r} = {}^{15}{C_{r}} 5^{r / 3} 2^{(3 - \frac{r}{5})}$

For rational terms, $\frac{r}{3}$ and $\frac{r}{5}$ must be an integer

$∴$ 3 and 5 divide $r \Rightarrow 15$ divides $r \Rightarrow r = 0$ and $r = 15$

Hence, ${}^{15}{C_{0}} 5^{0} 2^{3} + {}^{15}{C_{15}} 5^{5} 2^{0}$

$∴$ Required sum = 8 + 3125 = 3133

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