The sum of the rational terms of 21/5+320is - Infinity Learn by Sri Chaitanya

A
$71$
B
$85$
C
$97$
D
none of these

Solution:

$T_{r + 1,}$ the $(r + 1)$ th term in the expansion of ${(2^{1 / 5} + \sqrt{3})}^{20}$ is

$T_{r + 1} =^{20} C_{r} {(2^{1 / 5})}^{20 - r} {(3^{1 / 2})}^{r} =^{20} C_{r} 2^{4 - r / 5} (3)^{r / 2}$
As $2$ and $3$ are relatively prime $2^{4 - r / 5}$ and $3^{r / 2}$ will be rational if both $4 - r / 5$ and $r / 2$ are integers, i.e. if $r / 5$ and $r / 2$ are integers. This is possible if
$r = 0, 10, 20 .$
Thus, the sum of rational terms is
$^{20} C_{0} (2^{4}) (3^{0}) +^{20} C_{10} (2^{2}) (3^{5}) +^{20} C_{20} (2^{0}) (3^{10})$
As this is more than $71, 85$ or $97$ , answer is (4).

The sum of the rational terms of ${(2^{1 / 5} + \sqrt{3})}^{20}$ is

Solution:

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