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  • If the sum of the coefficients of all even powers of x in the product <img alt="(1+x+x^{2}+....+x^{2n})(1-x+x^{2}-x^{3}+....+x^{2m})" src="/latex-image/?%281&plus;x&plus;x%5E%7B2%7D&plus;....&plus;x%5E%7B2n%7D%29%281-x&plus;x%5

If the sum of the coefficients of all even powers of x in the product (1+x+x^{2}+....+x^{2n})(1-x+x^{2}-x^{3}+....+x^{2m}) is 61, then n is equal to _________.
Option: 1 30
Option: 260
Option: 315
Option: 4 45
 

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\text { Let } (1+x+x^{2}+....+x^{2n})(1-x+x^{2}-x^{3}+....+x^{2n})=a_{0}+a_{1} x+a_{2} x^{2}+\ldots \ldots

\\Put \,\, x=1 \\ (2 n+1).1 =a_{0} + a_{1} + a_{2}+ \ldots \ldots \\ Put\, x =-1 \\ 1.(2 n+1)= a_{0} - a_{1} + a_{2}+\ldots \ldots \\ From (i)+(ii) \\ 4 n+2 =2 (a_{0}+a_{2}+\ldots ) \\ So,\,\, 2 n+1=61 \\ \Rightarrow n=30

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Kuldeep Maurya

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