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(Solved): 17 By giving x appropriate numerical values in the expansion of (1+x)^(2n), where n is a positive in ...



17 By giving

x

appropriate numerical values in the expansion of

(1+x)^(2n)

, where

n

is a positive integer, prove that (a)

^(2n)C_(1)+^(2n)C_(2)+^(2n)C_(3)+cdots+^(2n)C_(2n-1)=2(2^(2n-1)-1)

(b)

1-2^(2n)C_(1)+4^(2n)C_(2)-8^(2n)C_(3)+cdots+(-2)^(2n)=1

(c)

^(2n)C_(0)+^(2n)C_(2)+^(2n)C_(4)+cdots+^(2n)C_(2n)=^(2n)C_(1)+^(2n)C_(3)+^(2n)C_(5)+cdots+^(2n)C_(2n-1)=2^(2n-1)


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