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

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 $$n = 7 + 8 + 9 + 10 + 11 + 12$$

$$ = 57$$

Binomial Theorem

Continue this paper