`If a + b = k, ` when `a, b gt o` and
`S(k, n) = sum _(r=0)^(n) r^(2) (""^(n) C_(r) ) a^(r) cdot b^(n-r) ` , then

Options

A. `S(1,3) =3 (3a^(2)+ab) `
B. `S(2,4)=16(4a^(2)+ab)`
C. `S(3,5) = 25 (5a^(2) + ab) `
D. `S(4,6)=36(6a^(2)+ab)`

Answer

a, b

Explanation

`because S(k,n) =sum_(r=0) ^(n) r^(2) cdot (""^(n) C_(r) ) a^(r) cdot b^(n-r)`
`=b^(n) sum_(r=0)^(n) r^(2) cdot (""^(n/r) cdot^(n-1)C_(r-1))cdot (a/b)^(r)`
`nb^(n)sum_(r=0)^(n) ((r-1) + 1) ^(n-1) C_(r-1)cdot (a/b)^(r)`
`nb^(n)sum_(r=0)^(n) ((n-1) cdot ^(n-1)C_(r-2)+^(n-1)C_(r-1)) (a/b)^(r)`
`=nb^(n) cdot (n-1) cdot (a/b)^(2) sum _(r=0)^(n) ""^(n-2)C_(r-2)(a/b)^(r-2)`
`+nb^(n)cdot (a/b)sum_(r=o)^(n) ""^(n-1) C_(r-) (a/b)^(r-1)`
`=nb^(n) cdot (n-1)(a/b)^2(1+a/b) ^(n-2) + nb^(n) cdot (a/b) .(1+a/b)^(n-1)`
`= n(n-1) a^(2)k^(n-2)+nak^(n-1)`
`=n^2a^(2)k^(n-2) + nak^(n-2) (k-a) = n^(2)a^(2) k^(n-2) + nabk^(n-2)` `therefore S(1,3) = 9a^(2) + 3ab = 3 (3a^(2) + ab) [because a + b = k]`
`S(2,4) = 16 (4a^(2) + ab)`
`S(3,4) = 135 (5a^(2) + ab)`
` S(4,6) = 1536 (6a^(2)+ab)`