If the sum of the coefficients of all the positive powers of x , in the binomial expansion of x n + 2 x 5 7 is 939 , then the sum of all the possible integral values of n is

Question

If the sum of the coefficients of all the positive powers of x , in the binomial expansion of x n + 2 x 5 7 is 939 , then the sum of all the possible integral values of n is,

MARKS App · Accepted Answer

Plotting the table for coefficient of x n + 2 x 5 7 and there cumulative sum : Coefficient Commulative sum x 7 n → C 0 7 1 x 6 n - 5 → 2 × C 1 7 1 + 14 x 5 n - 10 → 2 2 × C 2 7 1 + 14 + 84 x 4 n - 15 → 2 3 × C 3 7 1 + 14 + 84 + 280 x 3 n - 20 → 2 4 × C 4 7 1 + 4 + 84 + 280 + 560 = 939 x 2 n - 25 → 2 5 × C 5 7 So, 3 n - 20 ≥ 0 ∩ 2 n - 25 0 ∩ n ∈ I ∴ 7 ≤ n ≤ 12 Sum = 7 + 8 + 9 + 10 + 11 + 12 = 57

Coefficient	Commulative sum
$x^{7 n} \to {}^{7}C_{0}$	$1$
$x^{6 n - 5} \to 2 \times {}^{7}C_{1}$	$1 + 14$
$x^{5 n - 10} \to 2^{2} \times {}^{7}C_{2}$	$1 + 14 + 84$
$x^{4 n - 15} \to 2^{3} \times {}^{7}C_{3}$	$1 + 14 + 84 + 280$
$x^{3 n - 20} \to 2^{4} \times {}^{7}C_{4}$	$1 + 4 + 84 + 280 + 560 = 939$
$x^{2 n - 25} \to 2^{5} \times {}^{7}C_{5}$

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