Question

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

• A. 11
• B. 10
• C. 12
• D. 8

Answer 1

The correct option is A 11

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 =${}^n{C}_{r+2}=462$ .
$\therefore \frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{n-r}{r+1}=2$ $or,n-r=2(r+1)or,r=\frac{1}{3}(n-2)$ $and\frac{{}^n{C}_{r+2}}{{}^n{C}_{r+1}}=\frac{n-r-1}{r+2}=\frac{231}{165}$ $or165(n-r-1)=231(r+2)or165n-627=396r$
$or165n-627=396\times \frac{1}{3}\times (n-2)$
$or165n-627=132(n-2)orn=11$