The sum of all rational terms in the expansion of (2+ 3 )^8 is:

Question

Accepted Answer

Correct answer: B. (b) To find the sum of all rational terms in the expansion of (2+ 3 )^8, we consider the binomial expansion: S= ^8C_0 (2 )^8+ ^8C_1 (2 )^7 ( 3 )+ …+ ^8C_8 ( 3 )^8 We need to identify and sum only the rational terms. In the binomial expansion, a term is rational if the exponent of 3 is even. Thus, the rational terms are: ^8C_0 (2 )^8+ ^8C_2 (2 )^6 ( 3 )^2+ ^8C_4 (2 )^4 ( 3 )^4+ ^8C_6 (2 )^2 ( 3 )^6+ ^8C_8 ( 3 )^8 Evaluating these terms: ^8C_0 (2 )^8=256 ^8C_2 (2 )^6 (3 )= ^8C_2 64 3=1344 ^8C_4 (2 )^4 (3 )^2= ^8C_4 16 9=3024 ^8C_6 (2 )^2 (3 )^3= ^8C_6 4 27=4032 ^8C_8 (3 )^4=1 81=81 Summing these rational terms gives: 256+1344+3024+4032+81=18817 Therefore, the sum of all rational terms in the expansion is 18,817.

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

Solution

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