InterviewSolution
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InterviewSolution
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Show that the coefficient of the middle term in the expansion of `(1+x)^(2n)`is equal to the sum of the coefficients of two middle terms in the expansion of `(1+x)^(2n-1)`. |
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Answer» Middle terms in expansion `(1+x)^(2n-1)` are `T_n` and `T_(n+1).` So, sum of coefficients of these two middle terms will be, `C(2n-1,n-1)+C(2n-1,n) = ((2n-1)!)/((n-1)!n!) + ((2n-1)!)/(n!(n-1)!)` `=2((2n-1)!)/(n!(n-1)!) = (2n((2n-1)!))/(n(n-1)!(n!))` `=(2n!)/((n!)(n!))->(1)` Now, we will find the coefficient of middle term `T_(n+1)` in expansion `(1+x)^(2n)`. Coefficient of `T_(n+1)` in `(1+x)^(2n)` `= C(2n,n) = (2n!)/((n!)(n!))->(2)` From `(1)` and `(2)`, we can see that both values are equal. |
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