`sum_(r=0)^(300)a_r x^r=(1+x+x^2+x^3)^(100)dot` If `a=sum_(r=0)^(300)a_r ,t h e nsum_(r=0)^(300)r a_r` is equal to `300 a` b. `100 a` c. `150 a` d. `75 a`

Options

A. `.^(n)C_(r)`
B. `.^(n)C_(r)3^(r)`
C. `.^(2n)C_(r )`
D. `.^(n)C_(r )2^(r )`

Answer

D

Explanation

`(1-x)^(n)(1+x)^(n)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n)(1-x)^(n-r)`
or `(1-x+2x)^(n) = underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)`
or `underset(r=0)overset(n)sum.^(n)C_(r)(1-x)^(n-r)(2x)^(r)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)`
Comparing general term, we get `a_(r) = .^(n)C_(r)2^(r)`.