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If x + y = 1, prove that sum(r=0)^(n) r"...

If `x + y = 1`, prove that `sum_(r=0)^(n) r""^(n)C_(r) x^(r ) y^(n-r) = nx`.

Text Solution

Verified by Experts

We have
`underset(r=0)overset(n)sumr.^(n)C_(r)x^(r )y^(n-r) = underset(r=1)overset(n)sumn.^(n-1)C_(r-1)x^(r-1)x^(1)y^(n-r)`
` = nx underset(r=1)overset(n)sum .^(n-1)C_(r-1)x^(r-1)y^((n-1)-(r-1))`
`= nx(x+y)^(n-1)`
` = nx , [:' x + y = 1]`
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