The coefficients of three consecutive terms of ( 1 + x ) n + 5 are in the ratio 5 : 10 : 14 . Then n = [2013]

Question

The coefficients of three consecutive terms of ${(1 + x)}^{n + 5}$ are in the ratio $5 : 10 : 14$ . Then $n =$ [2013]

Smart Achievers · Accepted Answer

(6)

Let the coefficients of three consecutive terms of ${(1 + x)}^{n + 5}$ be ${}^{n + 5}C_{r - 1}, {}^{n + 5}C_{r}, {}^{n + 5}C_{r + 1}$ ,

then we have ${}^{n + 5}C_{r - 1} : {}^{n + 5}C_{r} : {}^{n + 5}C_{r + 1} = 5 : 10 : 14$

$\frac{{}^{n + 5}C_{r - 1}}{{}^{n + 5}C_{r}} = \frac{5}{10} \Rightarrow \frac{r}{n + 6 - r} = \frac{1}{2}$

$\Rightarrow n - 3 r + 6 = 0 . . . (i)$

$Also \frac{{}^{n + 5}C_{r}}{{}^{n + 5}C_{r + 1}} = \frac{10}{14} \Rightarrow \frac{r + 1}{n - r + 5} = \frac{5}{7}$

$\Rightarrow 5 n - 12 r + 18 = 0 . . . (i i)$

$Solving (i) and (ii), we get n = 6$

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