The sum of the series
`""^(3)C_(0)- ""^(4)C_(1) . (1)/(2) + ""^(5)C_(2)((1)/(2))^(2) - ""^(6)C_(3)((1)/(2))^(3) + ...` to `infty`, is

Options

A. 16
B. 8
C. `(16)/(81)`
D. none of these

Answer

c

Explanation

We have,
`(1 + x)^(-n) = ""^(n-1)C_(0) - ""^(n)C_(1) x + ""^(n-1)C_(2) x + ""^(n+1)C_(2) x^(2) - ""^(n+2)C_(3) x^(3) + ...`
On comparing the two series, we get
n-1 = 3 a d` x = (1)/(2) rArr n=4 and x = (1)/(2)`
Hence, sum of the given series = `(1 + (1)/(2))^(-4)` = (16)/(81)` .