The coefficients of three successive terms in | vedclass

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Question

(a) Let the coefficient of three consecutive terms i.e. ${(r + 1)^{th}},\,\,\,\,{(r + 2)^{th}},\,\,\,{(r + 3)^{th}}$ in expansion of ${(1 + x)^n}$ are

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(a) Let the coefficient of three consecutive terms i.e. ${(r + 1)^{th}},\,\,\,\,{(r + 2)^{th}},\,\,\,{(r + 3)^{th}}$ in expansion of ${(1 + x)^n}$ are 165,330 and 462 respectively then, coefficient of ${(r + 1)^{th}}$ term $ = {}^n{C_r} = 165$

Coefficient of $(r + 2)^{th}$ term $ = {}^n{C_{r + 1}} = 330$ and

Coefficient of $(r + 3)^{th}$ term $ = {}^n{C_{r + 2}} = 462$
 $\frac{{{}^n{C_{r + 1}}}}{{{}^n{C_r}}} = \frac{{n - r}}{{r + 1}} = 2$

or $n - r = 2(r + 1)$ or $r = \frac{1}{3}(n - 2)$

and $\frac{{{}^n{C_{r + 2}}}}{{{}^n{C_{r + 1}}}} = \frac{{n - r - 1}}{{r + 2}} = \frac{{231}}{{165}}$

or $165(n - r - 1) = 231(r + 2)$ or $165n - 627 = 396r$

or $165n - 627 = 396 	imes \frac{1}{3} 	imes (n - 2)$

or $165n - 627 = 132(n - 2)$ or $n = 11.$

The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be

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