Question

If some three consecutive coefficients in the binomial expansion of (x+ 1)ⁿ in powers of x are in the ratio 2:15:70, then the average of these three coefficients is ______.

Options

• 964

• 232

• 227

• 625

Answer 1

If some three consecutive coefficients in the binomial expansion of (x+ 1)ⁿ in powers of x are in the ratio 2:15:70, then the average of these three coefficients is 232.

Explanation:

Given ⁿC_r–1 : ⁿC_r : ⁿC_r+1 = 2 : 15 : 70

⇒ `(""^nC_(r-1))/(""^nC_r) = 2/15` and `(""^nC_r)/(""^nC_(r + 1)) = 15/70`

⇒ `r/(n - r + 1) = 2/15` and `(r + 1)/(n - r) = 3/14`

⇒ 17r = 2n + 2 and 17r = 3n – 14

i.e., 2n + 2 = 3n – 14

⇒ n = 16 and r = 2

∴ Average = `(""^16C_1 + ""^16C_2 + ""^16C_3)/3`

= `(16 + 120 + 560)/3`

= `696/3`

= 232