Question

If `(1 +x+x^2)^25 = a_0 + a_1x+ a_2x^2 +..... + a_50.x^50` then `a_0 + a_2 + a_4 + ... + a_50` is :
A. even
B. odd and of the form `3n`
C. odd and of the form `(3n-1)`
D. odd and of the form `(3n+1)`

Answer 1

Correct Answer - A
`(a)` Putting `x=1` and `-1` and adding
`a_(0)+a_(2)+…+a_(50)=(3^(25)+1)/(2)`
`=((1+2)^(25)+1)/(2)`
`=("^(25)C_(0)+^(25)C_(1)*2+^(25)C_(2)*2^(2)+^(25)C_(25)*2^(25)+1)/(2)`
`=(2[1+^(25)C_(1)+^(25)C_(2)*2+...+^(25)C_(25)*2^(24)])/(2)`
`=2[13+^(25)C_(2)+...+^(25)C_(25)*2^(23)]`