JEE Main 2022 · 27 Jun · Shift 2 · Mathematics Q24

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MathematicsInteger+4 / -1Question 24 of 90

If the sum of the coefficients of all the positive powers of $x$ , in the Binomial expansion of $\left(x^n+\dfrac2{x^5}\right)^7$ is 939, then the sum of all the possible integral values of $n$ is ______.

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Answer and worked solution

Official answer: 57

Term. $\binom7r2^rx^{n(7-r)-5r}$ .

Positive powers need $r<\dfrac{7n}{n+5}$ . The coefficients for $r=0,\dots,4$ are $1,14,84,280,560$ , summing to 939, so exactly $r\le4$ must qualify: $4<\frac{7n}{n+5}\le5$ , i.e. $\frac{20}3<n\le12.5$ , so $n=7,8,\dots,12$ .

Sum. $7+8+9+10+11+12=57$ .

Answer: 57

Topics:General and middle terms

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JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 24