First slide

Binomial theorem for positive integral Index

Question

If the coefficient of $r^{th} (r + 1)^{th}$ and $(r + 2)^{th}$ terms in the expansion of $(1 + x)^{14}$ are in A.P ., then the value of $r,$ is

Moderate

A

5,9
B

6,9
C

7,9
D

none of these

Solution

The coefficient of $f (r + 1)^{th}$ term in the expansion of $(1 + x)^{14}$ is $^{14} C_{r}$
It is given that
$^{14} C_{r - 1},^{14} C_{r},^{14} C_{r + 1}$ are in A.P.
$\begin{array}{l} \Rightarrow 2^{14} C_{r} =^{14} C_{r - 1} +^{14} C_{r + 1} \\ \Rightarrow 2 = \frac{^{14} C_{r - 1}}{^{14} C_{r}} + \frac{^{14} C_{r + 1}}{^{14} C_{r}} \\ \Rightarrow 2 = \frac{r}{15 - r} + \frac{14 - r}{r + 1} \\ \Rightarrow 2 (15 - r) (r + 1) = r^{2} + r + 210 - 29 r + r^{2} \\ \Rightarrow 4 r^{2} - 56 r + 180 = 0 \\ \Rightarrow r^{2} - 14 r + 45 = 0 \Rightarrow r = 5, 9 . \end{array}$

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