if-a-1-a-2-a-3-a-4-are-the-coefficient-of-any-four-four-consecutive-terms-in-the-expansion-of-1-x-n-then-a-1-a-2-a-1-a-3-a-3-a-4-is-equal-to-

Question Number 63858 by gunawan last updated on 10/Jul/19

if a_{1}, a_{2}, a_{3}, a_{4} are the coefficient

of any four four consecutive

terms in the expansion of {(1 + x)}^{n}

then \frac{a_{1}}{a_{2} + a_{1}} + \frac{a_{3}}{a_{3} + a_{4}} is equal to \dots

Answered by ajfour last updated on 10/Jul/19

v = \frac{^{n} C_{r - 1}}{^{n} C_{r} +^{n} C_{r - 1}} + \frac{^{n} C_{r + 1}}{^{n} C_{r + 1} +^{n} C_{r + 2}}

&^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}

\Rightarrow

v = \underset{r = r}{\sum^{r + 1}} \frac{n! r! (n - r + 1)!}{(r - 1)! (n - r + 1)! (n + 1)!}

= \frac{r}{n + 1} + \frac{r + 1}{n + 1} = \frac{2 r + 1}{n + 1} .

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