If the sum of the coefficient of all the positive even power

Question

If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of ( 2x^3 + 3 / x )^10 is 5^10 - beta . 3^9, then beta is equal to ____________.

DivineJEE · Accepted Answer

Given, Binomial Expansion ( 2x^3 + 3 / x )^10 General term T_r + 1 = ^10C_r . (2x^3)^10 - r . ( 3 / x )^r = ^10C_r . 2^10 - r . 3^r . x^30 - 3r . x^ - r = ^10C_r . 2^10 - r . 3^r . x^30 - 4r For positive even power of x, 30 - 4r should be even and positive. For r = 0, 30 - 4 x 0 = 30 (even and positive) For r = 1, 30 - 4 x 1 = 26 (even and positive) For r = 2, 30 - 4 x 2 = 22 (even and positive) For r = 3, 30 - 4 x 3 = 18 (even and positive) For r = 4, 30 - 4 x 4 = 14 (even and positive) For r = 5, 30 - 4 x 5 = 10 (even and positive) For r = 6, 30 - 4 x 6 = 6 (even and positive) For r = 7, 30 - 4 x 7 = 2 (even and positive) For r = 8, 30 - 4 x 8 = -2 (even but not positive) So, for r = 1, 2, 3, 4, 5, 6 and 7 we can get positive even power of x. Therefore Sum of coefficient for positive even power of x = ^10C_0 . 2^10 . 3^0 + ^10C_1 . 2^9 . 3^1 + ^10C_2 . 2^8 . 3^2 + \ ^10C_3 . 2^7 . 3^3 + ^10C_4 . 2^6 . 3^4 + ^10C_5 . 2^5 . 3^5 + \ ^10C_6 . 2^4 . 3^6 + ^10C_7 . 2^3 . 3^7 = ^10C_10 . 2^10 . 3^0 + ^10C_1 . 2^9 . 3^1 + ..... + \ ^10C_10 . 2^0 . 3^10 - \ [ ^10C_8 . 2^2 . 3^8 + ^10C_9 . 2 . 3^9 + ^10C_10 . 2^0 . 3^10 ] = (2 + 3)^10 - [ 45 . 4 . 3^8 + 10 . 2 . 3^9 + 1 . 1 . 3^10 ] = 5^10 - [ 60 x 3^9 + 20 . 3^9 + 3 . 3^9 ] = 5^10 - ( 60 + 20 + 3 )3^9 = 5^10 - 83 . 3^9 Therefore beta = 83

Explanation

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