(D) The general term in the expansion of $(2+\sqrt{3})^8$ is given by $T_{r+1} = { }^8 C_r (2)^{8-r} (\sqrt{3})^r$. For the term to be rational,$r$ mu

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(D) The general term in the expansion of $(2+\sqrt{3})^8$ is given by $T_{r+1} = { }^8 C_r (2)^{8-r} (\sqrt{3})^r$. For the term to be rational,$r$ must be an even integer,i.e.,$r \in \{0, 2, 4, 6, 8\}$. The sum of rational terms is: $S = { }^8 C_0 (2)^8 + { }^8 C_2 (2)^6 (\sqrt{3})^2 + { }^8 C_4 (2)^4 (\sqrt{3})^4 + { }^8 C_6 (2)^2 (\sqrt{3})^6 + { }^8 C_8 (\sqrt{3})^8$ $S = 1 \cdot 256 + 28 \cdot 64 \cdot 3 + 70 \cdot 16 \cdot 9 + 28 \cdot 4 \cdot 27 + 1 \cdot 81$ $S = 256 + 5376 + 10080 + 3024 + 81$ $S = 18817$

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is

JEE MAIN 2025 All JEE MAIN

A
$16923$
B
$3763$
C
$33845$
D
$18817$

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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