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

Correct Answer: 57

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

\(7C_0 \cdot 2^0 + 7C_1 \cdot 2^1 + 7C_2 \cdot 2^2 + 7C_3 \cdot 2^3 + 7C_4 \cdot 2^4 = 939\)
\(∴ r = 4\)
\(∵ 7\ n–nr–5r = 0\)
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 \(n=57\)