First slide

Binomial theorem for positive integral Index

Question

If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)^{n + 5} are in the ratio 5 : 10 : 14, then the
Iargest coefficient in this expansion is

Moderate

A

330
B

462
C

792
D

252

Solution

Let the three consecutive terms in the binomial expansion of $(1 + x)^{n + 5}$ are
$^{n + 5} C_{r - 1},^{n + 5} C_{r} {and}^{n + 5} C_{r + 1}$
Now, according to the given information $^{n + 5} C_{r - 1} :^{n + 5} C_{r} :^{n + 5} C_{r + 1} = 5 : 10 : 14$
$\begin{array}{r} \Rightarrow \frac{(n + 5)!}{(r - 1)! (n - r + 6)!} : \frac{(n + 5)!}{r! (n - r + 5)!} : \frac{(n + 5)!}{(r + 1)! (n - r + 4)!} = 5 : 10 : 14 \end{array}$

$\begin{array}{l} \begin{aligned} \Rightarrow \frac{1}{(n - r + 6) (n - r + 5)} : \frac{1}{r (n - r + 5)} : \frac{1}{(r + 1) r} = 5 : 10 : 14 \\ So, \frac{r}{n - r + 6} = \frac{5}{10} \\ \Rightarrow 2 r = n - r + 6 \Rightarrow n + 6 = 3 r \\ and \frac{r + 1}{n - r + 5} = \frac{5}{7} \end{aligned} \\ \begin{array}{lrl} \Rightarrow & 7 r + 7 & = 5 n - 5 r + 25 \\ \Rightarrow & 5 n + 18 & = 12 r \end{array} \end{array}$
From Eqs. (i) and (ii), we have n =6.
So, the largest coefficient in the expansion is same as
the greatest binomial coefficient
$\begin{array}{l} =^{11} C_{5} {or}^{11} C_{6} \\ = \frac{11!}{5! 6!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2} = 462 \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App