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

$(r+1{)}^n$ 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

∴ $\frac{{}^n{C}_{r+1}}{{}^n{C}_r}$ = 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 × $\frac{1}{3}$ × (n - 2)

or 165n - 627 = 132(n - 2) or n = 11.