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

Given coefficients of( r-1)th, rth and (r+1)th terms in the expansion of (x+1)n are in the ratio 1:3:5
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)
C(n, r-2) : C(n, r-1) : C(n, r) = 1:3:5
Divide 1^st and 2^nd

C(n,r-2)/C(n,r-1) = 1 / 3

= $\left[\frac{n!}{(r-2)!(n-r+2)!}\right]\ast \left[\frac{(r-1)!(n-r+1)!}{n!}\right]$ = 1/3

$\left[\frac{(r-1)!(r-2)!}{(r-2)!(n-r+2)!}\right]$ = $\left[\frac{1}{3}\right]$

C(n,r-1)/C(n,r) = 3/5( r -1) / ( n -r +2) = 1 / 3
3r - 3 = n - r + 2
n - 4r = -5 ----------(1)
Divide 2nd and 3^rd

[n! / (r-1)! (n -r +1)! ] x r!(n-r)!/ n! = 3 / 5

r / (n -r + 1) = 3 / 5

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