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Consider a G.P. with first term (1+x)^(n), |x| lt 1, common ratio (1+x)/(2) and number of terms (n+1). Let 'S' be sum of all the terms of the G.P., then The coefficient of x^(n) is 'S' is |
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Answer» `2^(n)` `=(1)/(2^(n))((1+x)^(n)[2^(n+1)-(1+x)^(n+1)])/(1-x)` `=(1)/(2^(n))[2^(n+1)(1+x)^(n)-(1+x)^(2n+1)](1-x)^(-1)` `=(1)/(2^(n))[2^(n+1)(1+x)^(n)-(1+x)^(2n+1)](1+x+x^(2)+...+oo)` The COEFFICIENT of `x^(n)` in `S` `=(1)/(2^(n))[2^(n+1)sum_(r=0)^(n)'^(n)C_(r )-sum_(r=0)^(n)'^(2n+1)C_(r )]` `=(1)/(2^(n))[2^(n+1)2^(n)-(1)/(2)2^(2n+1)]` `=2^(n)` |
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