First slide

Binomial theorem for positive integral Index

Question

If coefficient of 9th, 10th and 11th terms of ${(1 + x)}^{n}$ are in $A . P .,$ then value of $n$ can be

Moderate

A

15
B

32
C

23
D

17

Solution

$\begin{array}{l} 2 (^{n} C_{9}) =^{n} C_{8} +^{n} C_{10} \\ \Rightarrow 2 = \frac{9}{n - 8} + \frac{n - 9}{10} \\ \Rightarrow 20 (n - 8) = 90 + (n - 8) (n - 9) \\ \Rightarrow n^{2} - 37 n + 322 = 0 \end{array}$
$\Rightarrow (n - 14) (n - 23) = 0$
Thus, $n$ can be 23

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