Question

The coefficients of the (r – 1)^th, r^th and (r + 1)^th terms in the expansion of

(x + 1)ⁿ are in the ratio 1:3:5. Find n and r.

Answer 1

It is known that (k + 1)^th term, (T_k+1), in the binomial expansion of (a + b)ⁿ is given by .

Therefore, (r – 1)^th term in the expansion of (x + 1)ⁿ is

r ^th term in the expansion of (x + 1)ⁿ is

(r + 1)^th term in the expansion of (x + 1)ⁿ is

Therefore, the coefficients of the (r – 1)^th, r^th, and (r + 1)^th terms in the expansion of (x + 1)ⁿ are respectively. Since these coefficients are in the ratio 1:3:5, we obtain

Multiplying (1) by 3 and subtracting it from (2), we obtain

4r – 12 = 0

⇒ r = 3

Putting the value of r in (1), we obtain

n – 12 + 5 = 0

⇒ n = 7

Thus, n = 7 and r = 3