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Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1). |
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Answer» SOLUTION :`S=UNDERSET(r=0)OVERSET(2N)sumr.(.^(2n)C_(r))^(2)` `= underset(r=0)overset(2n)sum(r..^(2n)C_(r))(.^(2n)C_(r))` `= underset(r=0)overset(2n)sum(2n)^(2n-1)C_(r-1)..^(2n)C_(2n-r)` `= 2n`(Coefficient of `x^(2n-1)` in the expansion of `(1+x)^(2n-1)(1+x)^(2n))` `= 2n`(coefficient of `x^(2n-1)` in the expansion of `(1+x)^(4n-1)`) `= 2n xx .^(4n-1)C_(2n-1)` |
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