If x+y=1, then Σ_(r=0) ^n r ^n C_r x^r y^(n−r) equals


Question and Answers Forum

All Questions Topic List
UNKNOWN Questions
Previous in All Question Next in All Question
Previous in UNKNOWN Next in UNKNOWN
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

$x + y = 1 \Rightarrow {(x + y)}^{n} = 1 \Rightarrow \sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} = 1 \Rightarrow$ $y^{n} + \sum_{r = 1}^{n} C_{n}^{r} x^{r} y^{n - r} = 1 l e t d e r i v a t e (y f i x e d) \Rightarrow$ $\sum_{r = 1}^{n} C_{n}^{r} r x^{r - 1} y^{n - r} = 0 \Rightarrow \frac{1}{x} \sum_{r = 1}^{n} C_{n}^{r} x^{r} y^{n - r} = 0 (x \neq 0) \Rightarrow$ $\sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} - y^{n} = 0 \Rightarrow \sum_{r = 0}^{n} C_{n}^{r} x^{r} y^{n - r} = y^{n}$ $i f x = 0 \Rightarrow y = 1 a n d \sum_{r = 0}^{n} (. . .) = 0$
	Answered by mr W last updated on 11/Jul/19

	${(x + y)}^{n} = 1$ $\underset{r = 0}{\sum^{n}} C_{r}^{n} x^{r} y^{n - r} = 1$ $\underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r - 1} y^{n - r} = 0$ $\underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r} y^{n - r} = 0 \times x$ $\Rightarrow \underset{r = 0}{\sum^{n}} {r C}_{r}^{n} x^{r} y^{n - r} = 0$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum