The sum of coefficient of integral powers of x in the binomial expansion of `(1-2sqrt(x))^(50)` is

Options

A. `(1)/(2) (3^(50) + 1)`
B. `(1)/(2) (3^(50))`
C. `(1)/(2) (3^(50) - 1)`
D. `(1)/(2) (3^(50) + 1)`

Explanation

Let `T_(t + 1)` be the general term in the expansion of `(1 - 2 sqrt(x))^(50)`
`:. T_(r + 1) = .^(50)C_(r ) (1)^(50 - r) (-2x^(1//2))^(r ) = .^(50)C_(r ) 2^(r ) x^(r//2) (-1)^(r )`
For the integral power of x,r should be even integer.
`:.` Sum of coefficientss `= underset(r = 0)overset(25)(sum) .^(50)C_(2r) (2)^(2r)`
`= (1)/(2) [(1 + 2)^(50) + (1 - 2)^(50)] = (1)/(2) (30^(50) + 1)`
Alternate solution
we have
`(1 - 2 sqrt(2))^(50) = C_(0) - C_(1) 2 sqrt(x) + C_(2) (sqrt(2x))^(2 + ... + C_(50) (2 sqrt(x))^(50)` .... (i)
`(1 + 2 sqrt(x))^(50) = C_(0) + C_(1) 2 sqrt(x) + C_(2) (2 sqrt(x))^(2) + ... + C_(50) (2 sqrt(x))^(50)` ..... (iii)
On adding Eqs. (i) and (ii) we get
`(1 - 2sqrt(x))^(50) + (1 + 2 sqrt(x))^(50)`
`= 2 [C_(0) + C_(2) (2 sqrt(x))^(2) + ... + C_(50) (2 sqrt(x))^(50)]`
`implies ((1 - 2 sqrt(x))^(50) + (1 + 2 sqrt(1))^(50))/(2)`
`= C_(0) + C_(2) (2 sqrt(x))^(2) + .... + C_(50) (2 sqrt(x))^(50)`
On putting x - 1, we get
`((1 - 2 sqrt(1))^(50) + (1 + 2 sqrt(1))^(50))/(2) = C_(0) + C_(2) + ... + C_(50) (2)^(50)`
`implies ((-1)^(50) + (3)^(50))/(2) = C_(0) + C_(2) (2)^(2) + ... + C_(50) (2)^(50)`
`implies (1 + 3^(50))/(2) = C_(0) + C_(2) (2)^(2) +... + C_(50) (2)^(50)` .