If-x-y-1-then-r-0-n-r-n-C-r-x-r-y-n-r-equals-

Question Number 63943 by gunawan last updated on 11/Jul/19

If x + y = 1, then \underset{r = 0}{\sum^{n}} r^{n} C_{r} x^{r} y^{n - r} equals

Commented by mathmax by abdo last updated on 11/Jul/19

x + y = 1 \Rightarrow {(x + y)}^{n} = 1 \Rightarrow \sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} = 1 \Rightarrow

y^{n} + \sum_{r = 1}^{n} C_{n}^{r} x^{r} y^{n - r} = 1 l e t d e r i v a t e (y f i x e d) \Rightarrow

\sum_{r = 1}^{n} C_{n}^{r} r x^{r - 1} y^{n - r} = 0 \Rightarrow \frac{1}{x} \sum_{r = 1}^{n} C_{n}^{r} x^{r} y^{n - r} = 0 (x \neq 0) \Rightarrow

\sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} - y^{n} = 0 \Rightarrow \sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} = y^{n}

i f x = 0 \Rightarrow y = 1 a n d \sum_{r = 0}^{n} (\dots) = 0

Answered by mr W last updated on 11/Jul/19

{(x + y)}^{n} = 1

\underset{r = 0}{\sum^{n}} C_{r}^{n} x^{r} y^{n - r} = 1

\underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r - 1} y^{n - r} = 0

\underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r} y^{n - r} = 0 \times x

\Rightarrow \underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r} y^{n - r} = 0

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