If `(1+x^(2))^(n) =sum_(r=0)^(n) a_(r)x^(r )= (1+x+x^(2)+x^(3))^(100)`. If `a = sum_(r=0)^(300)a_(r)`, then `sum_(r=0)^(300) ra_(r)` is

Monjur Alahi

If `(1+x^(2))^(n) =sum_(r=0)^(n) a_(r)x^(r )= (1+x+x^(2)+x^(3))^(100)`. If `a = sum_(r=0)^(300)a_(r)`, then `sum_(r=0)^(300) ra_(r)` is
A. `.^(n)C_(r)`
B. `.^(n)C_(r)3^(r)`
C. `.^(2n)C_(r )`
D. `.^(n)C_(r )2^(r )`


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Salim Ahmed Kawsar

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Correct Answer - D
`(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)`.

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