The sum of all rational terms in the expansion of is equal to [2025]

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Solution

Q.
The sum of all rational terms in the expansion of ${(2 + \sqrt{3})}^{8}$ is equal to [2025]

1 3763

2 18817

3 16923

4 33845

Ans.
(2)

We have,

${(2 + \sqrt{3})}^{8} = {}^{8}C_{0} 2^{8} + {}^{8}C_{1} 2^{7} (\sqrt{3}) + . . . + {}^{8}C_{8} {(\sqrt{3})}^{8}$

But, for rational terms, we take only those terms whose exponent is an even number.

Required sum = ${}^{8}C_{0} {(2)}^{8} + {}^{8}C_{2} {(2)}^{6} \cdot {(\sqrt{3})}^{2} + {}^{8}C_{4} {(2)}^{4} {(\sqrt{3})}^{4} + {}^{8}C_{6} {(2)}^{2} {(\sqrt{3})}^{6} + {}^{8}C_{8} {(\sqrt{3})}^{8}$

= $2^{8} + 28 \times 2^{6} \times 3 + 70 \times 2^{4} \times 9 + 28 \times 2^{2} \times 27 + 81$

= 256 + 5376 + 10080 + 3024 + 81 = 18817.

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