If the coefficient of the $r^{t h}, (r + 1)^{t h}$ & $(r + 2)^{t h}$ terms in the expansion of $(1 + x)^{14}$ are in $A P$ , find $r$ .

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Solution

$\begin{matrix} T_{r} =^{14} C_{r - 1} x^{15 - r} [^{n} C_{r} x^{h - r}] T_{r + 1} =^{14} C_{r + 1}^{14 - r} T_{r + 2} =^{14} C_{r + 1} x^{13 - r} c o f f i c i e n t a r e i n A . P 2 B = A + C 2^{14} C_{r} =^{14} C_{r - 1} +^{14} C_{r + 1} \frac{214!}{(14 - r)! r!} = \frac{14!}{(13 - R)! (R + 1)!} + \frac{14!}{(15 - r)! (r - 1)!} \frac{2}{(14 - r)! r!} = \frac{(15 - r)! (r - 1!)}{(13 - r)! (r + 1)!} + \frac{(13 - r)! (r - 1)!}{(15 - r)! (r - 1)!} 2 (15 - r) (r + 1) = (5 - r) (14 - r) + r (r + 1) 2 (- r^{2} + 14 r + 15) = 210 - 29 r + r^{2} + r^{2} + r 4 r^{2} - 56 r + 180 = 0 r^{2} - 14 r + 45 = 0 (r - 9) (r - 5) = 0 r = 5, 9 \end{matrix}$

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