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 _________.

$Given, Binomial expression is \ = ( x^n + (2 / x^5) )^7 \ Therefore General term \ T_r + 1 = ^7C_r * (x^n)^7 - r * ( (2 / x^5) )^r \ = ^7C_r * x^7n - nr - 5r * 2^r \ For positive power of x, \ 7n - nr - 5r > 0 \ => 7n > r(n + 5) \ => r 20 \ => n > (20 / 3) => n > 6.66 \ and \ 7n le 5n + 25 => 2n le 25 \ => n le 12.5 \ Therefore 6.66

If the sum of the coefficients of all the positive powe... | null (Mathematics)

\text{Given, Binomial expression is} \\ = \left( x^n + \frac{2}{x^5} \right)^7 \\~\\ \therefore \text{General term} \\ T_{r + 1} = {}^7C_r \cdot (x^n)^{7 - r} \cdot \left( \frac{2}{x^5} \right)^r \\ = {}^7C_r \cdot x^{7n - nr - 5r} \cdot 2^r \\~\\ \text{For positive power of } x, \\ 7n - nr - 5r > 0 \\ \Rightarrow 7n > r(n + 5) \\ \Rightarrow r < \frac{7n}{n + 5} \\~\\ \text{As } r \text{ represents the term of the binomial expression, } \\ r \text{ is always an integer.} \\~\\ \text{Given that the sum of coefficients is 939.} \\~\\ \text{When } r = 0, \\ \text{Sum of coefficients} = {}^7C_0 \cdot 2^0 = 1 \\~\\ \text{When } r = 1, \\ \text{Sum of coefficients} = {}^7C_0 \cdot 2^0 + {}^7C_1 \cdot 2^1 = 1 + 14 = 15 \\~\\ \text{When } r = 2, \\ \text{Sum of coefficients} = {}^7C_0 \cdot 2^0 + {}^7C_1 \cdot 2^1 + {}^7C_2 \cdot 2^2 \\ = 1 + 14 + 84 = 99 \\~\\ \text{When } r = 3, \\ \text{Sum of coefficients} = {}^7C_0 \cdot 2^0 + {}^7C_1 \cdot 2^1 + {}^7C_2 \cdot 2^2 + {}^7C_3 \cdot 2^3 \\ = 1 + 14 + 84 + 280 = 379 \\~\\ \text{When } r = 4, \\ \text{Sum of coefficients} = {}^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 \\ = 1 + 14 + 84 + 280 + 560 = 939 \\~\\ \therefore \text{For } r = 4, \text{ sum of coefficients } = 939 \\~\\ \text{To get the value of } r = 4, \text{ the value of } \frac{7n}{n + 5} \text{ should be between 4 and 5.} \\~\\ \therefore 4 < \frac{7n}{n + 5} \le 5 \\ \Rightarrow 4n + 20 < 7n \le 5n + 25 \\~\\ \therefore 4n + 20 < 7n \Rightarrow 3n > 20 \\ \Rightarrow n > \frac{20}{3} \Rightarrow n > 6.66 \\~\\ \text{and} \\ 7n \le 5n + 25 \Rightarrow 2n \le 25 \\ \Rightarrow n \le 12.5 \\~\\ \therefore 6.66 < n \le 12.5 \\~\\ \therefore \text{Possible integer values of } n = 7, 8, 9, 10, 11, 12 \\~\\ \therefore \text{Sum of values of } n = 7 + 8 + 9 + 10 + 11 + 12 \\ = 57

Explanation

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