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 both n and r.

Coefficient of (r-1)th term = C(n, r-2)

Coefficient of rth term = C(n, r-1)

Coefficient of (r+1)th term = C(n, r)

Considering 1^st and ^2nd

 

3r - 3 = n - r + 2

n - 4r = -5 ----------(1)

Considering 2nd and 3rd

 

5r = 3n - 3r +3

3n - 8r = -3 ---------(2)

2(n - 4r = -5)

2n - 8r = -10 ---------(3) 

Subtract (3) from (2)

n = 7

Substitute n = 7 in (2)

We get r = 3

n = 7, r = 3