Question

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of $\left(x^n+\frac{2}{x^5}\right)^7$ is 939, then the sum of all the possible integral values of n is ____________.

Answer 1

\left(x^n+\frac{2}{x^5}\right)^7$$=$$\sum_{r=0}^7 ^7C_r$$(x^n)^{7-r} \cdot \left(\frac{2}{x^5}\right)^r$$=$$\sum_{r=0}^7 $^7C_r$⋅$2^r \cdot x^{7n - nr - 5r}$

$7C_0 \cdot 2^0 + 7C_1 \cdot 2^1 + 7C_2 \cdot 2^2 + 7C_3 \cdot 2^3 + 7C_4 \cdot 2^4 = 939$
$∴ r = 4$
$∵ 7\ n–nr–5r = 0$
and r = 4 then
$n>\frac{20}{3}$
and r should not be 5
$∴n<\frac{25}{2}$
$∴$ Possible values of n are $7, 8, 9, 10, 11, 12$
$∴$ Sum of integral value of $n=57$