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

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Accepted Answer

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Given, Binomial expression is

$= {\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$

$\therefore$ General term

$${T_{r + 1}} = {}^7{C_r}\,.\,{({x^n})^{7 - r}}\,.\,{\left( {{2 \over {{x^5}}}} \right)^r}$$

$= {}^7{C_r}\,.\,{x^{7n - nr - 5r}}\,.\,{2^r}$

For positive power of x,

$7n - nr - 5r > 0$

$\Rightarrow 7n > r(n + 5)$

$\Rightarrow r < {{7n} \over {n + 5}}$

As r represent term of binomial expression so r is always integer.

Given that sum of coefficient is 939.

When $r = 0$,

sum of coefficient $= {}^7{C_0}\,.\,{2^0} = 1$

when $r = 1$,

sum of coefficient $= {}^7{C_0}\,.\,{2^0} + {}^7{C_1}\,.\,{2^1} = 1 + 14 = 15$

when $r = 2$,

sum of coefficient

$= {}^7{C_0}\,.\,{2^0} + {}^7{C_1}\,.\,{2^1} + {}^7{C_2}\,.\,{2^2}$

$= 1 + 14 + 84$

$= 99$

when $r = 3$,

sum of coefficient

$$ = {}^7{C_0}\,.\,{2^0} + {}^7{C_1}\,.\,{2^1} + {}^7{C_2}\,.\,{2^2} + {}^7{C_3}\,.\,{2^3}$$

$= 1 + 14 + 84 + 280$

$= 379$

when $r = 4$,

sum of coefficient

$$ = {}^7{C_0}\,.\,{2^0} + {}^7{C_1}\,.\,{2^1} + {}^7{C_2}\,.\,{2^2} + {}^7{C_3}\,.\,{2^3} + {}^7{C_4}\,.\,{2^4}$$

$= 1 + 14 + 84 + 280 + 560$

$= 939$

$\therefore$ For r = 4 sum of coefficient = 939

To get value of r = 4, value of ${{7n} \over {n + 5}}$ should be between 4 and 5.

$\therefore$ $4 < {{7n} \over {n + 5}} < 5$

$\Rightarrow 4n + 20 < 7n < 5n + 25$

$\therefore$ $4n + 20 < 7n$

$\Rightarrow 3n > 20$

$\Rightarrow n > {{20} \over 3}$

$\Rightarrow n > 6.66$

and

$7n < 5n + 25$

$\Rightarrow 2n < 25$

$\Rightarrow n < 12.5$

$\therefore$ $6.66 < n < 12.5$

$\therefore$ Possible integer values of $n = 7,8,9,10,11,12$

$\therefore$ Sum of values of $n = 7 + 8 + 9 + 10 + 11 + 12$

$= 57$

"

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

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

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