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Prove that (1^2)/3""^n C1+(1^2+2^2)/5^n ...

Prove that `(1^2)/3""^n C_1+(1^2+2^2)/5^n C_2(1^1+2^2+3^2)/7^n C_3+` `+(1^2+2^2++n^2)/(2n+1)^n C_n=(n(n+3))/62^(n-2)dot`

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`underset(r=0)overset(n)sumr^(2).^(n)C_(r)p^(r)q^(n-r)`
`= underset(r=0)overset(n)sumnr.^(n-1)C_(r-1)S=underset(r=1)overset(n)sum(1^(2)+2^(2)+"...."+r^(3))/(2r+1).^(n)C_(r)`
`= underset(r=1)overset(n)sum(r(r+1)(2r+1))/(6(2r+1)).^(n)C_(r)`
`= 1/6underset(r=1)overset(n)sumr(r+1).^(n)C_(r)`
`= 1/6underset(r=1)overset(n)sum(r+1).n..^(n-1)C_(r-1)`
`=1/6n underset(r=1)overset(n)sum((r-1)+2)^(n-1)C_(r-1)`
`=1/6n.underset(r=1)overset(n)sum((r-1)..^(n-1)C_(r-1)+2..^(n-1)C_(r-1))`
`= 1/6n.underset(r=1)overset(n)sum((n-1)..^(n-2)C_(r-2)+2..^(n-1)C_(r-1))`
`=1/6n.(n-1).2^(n-2)+(n)/(3).2^(n-1)=1/6n(n+3)2^(n-2)`
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