Question

Find the 6^th term of the expansion (y^1/2 + x^1/3)ⁿ , if the binomial coefficient of the 3^rd term from the end is 45.

Answer 1

Given : 3^rd term from the end =45 

To Find : 6^th term 

For (y^1/2 + x^1/3) ⁿ , 

a = y^1/2, b = x^1/3

We have, t_r+1 = (ⁿ_r)a^n-rb^r

As n = n, therefore there will be total (n+1) terms in the expansion. 

3^rd term from the end = (n+1-3+1)^th i.e. (n-1)^th term from the starting

For (n-1)^th term, r = (n-1-1) = (n-2)

t_(n-1) = t_(n-2)+1

Therefore 3^rd term from the end = \(\frac{n(n-1)}{2}\) (Y)(X)\(\frac{n-2}{3}\)

Therefore coefficient 3^rd term from the end = \(\frac{n(n-1)}{2}\) 

45 =   \(\frac{n(n-1)}{2}\) 

•90 = n (n-1) 

• 10 (9) = n (n-1) 

Comparing both sides, n = 10 

For 6^th term, r = 5

t₆ = t₅₊₁