Given `(1-x^(3))^(n)=sum_(k=0)^(n)a_(k)x^(k)(1-x)^(3n-2k)` then the value of `3*a_(k-1)+a_(k)` is (a) `"^(n)C_(k)*3^(k)` (b) `"^(n+1)C_(k)*3^(k)` (c) `"^(n+1)C_(k)*3^(k-1)` (d) `"^(n)C_(k-1)*3^(k)`

Options

A. `"^(n)C_(k)*3^(k)`
B. `"^(n+1)C_(k)*3^(k)`
C. `"^(n+1)C_(k)*3^(k-1)`
D. `"^(n)C_(k-1)*3^(k)`

Answer

B

Explanation

`(b)` Given `{(1-x)(1+x+x^(2))}^(n)`
`=sum_(k=0)^(n)a_(k)x^(k)((1-x)^(3n))/((1-x)^(2k))`
`{(1+x+x^(2))/((1-x)^(2))}=sum_(k=0)^(n)a_(k)(x^(k))/((1-x)^(2k))`
`{((1-x)^(2)+3x)/((1-x)^(2))}^(n)=sum_(k=0)^(n)a_(k){(x)/((1-x)^(2))}^(k)`
`{1+(3x)/((1-x)^(2))}^(n)=sum_(k=0)^(n)a_(k){(x)/((1-x)^(2))}^(k)`
Put `(x)/((1-x)^(2))=t`
`(1+3t)^(n)=sum_(k=0)^(n)a_(k)t^(k)`
`3*a_(k-1)+a_(k)=3("coefficient of"t^(k-1))+"coefficient of" t^(k)`
`=3.3^(k-1)'^(n)C_(k-1)+3^(k)'^(n)C_(k)`
`=3^(k)("^(n)C_(k-1)+^(n)C_(k))`
`=3^(k)('^(n+1)C_(k))`