Question

The sum of coefficients in integral powers of x in the binominal expansion `(1 - 2 sqrt(x))^(50)` is
A. `(1)/(2)(3^(50) -1)`
B. `(1)/(2) (2^(50) +1)`
C. `(1)/(2) (3^(50) +1)`
D. `(1)/(2) (3^(50))`

Answer 1

Correct Answer - c
We observe that only the odd terms in the
binomial expansion of `(1-2sqrt(x))^(50)` contain integral powers of x.
Also,
`(1 + 2 sqrt(x))^(50) + (1 - 2sqrt(x))^(50)`
`= 2 {""^(50)C_(0) + ""^(50)C_(1) (2 sqrt(x))^(2) + ""^(50)C_(4) (2sqrt(x))^(4) + ... +""^(50)C_(50)(2 sqrt(x))^(50)}` Replacing c by 1 on both side, we obtain
`3^(50) + (-1)^(50) = 2 {""^(50)C_(0) + ""^(50)C_(1) (2)^(2) + ""^(50)C_(4) (2)^(4) + ... +""^(50)C_(50)(2)^(50)}`
`rArr (1)/(2) (3^(50) +1)` = Sum of the coefficeints of integral power of x.