Find the 4th term from the beginning and end in the expansion of (3√2 + 1/3√3)^n

Question

Find the 4^th term from the beginning and end in the expansion of \(\Big(3\sqrt2 + \frac{1}{3\sqrt3}\Big)^n\)

Answer 1

To Find : 

I. 4^th term from the beginning 

II. 4^th term from the end

Formulae :

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

Therefore,

I. For the 4^th term from the starting. 

We have a formula,

Therefore, a 4^th term from the starting = \(\frac{n!}{(n-3)!\times3!}.\frac{(2)\frac{n-3}{3}}{3}\) 

Now,

II. For the 4^th term from the end 

We have a formula,

t_r+1 = (_rⁿ) a^n-rb^r

For t_(n-2) , r = (n-2)-1 = (n-3)

t_(n-2) = t_(n-3)+1

Therefore, a 4th term from the end =  \(\frac{n!}{(n-4)!\times4!}.(2)(3)\frac{3-n}{3}\) 

Conclusion :

I. 4th term from the beginning =   \(\frac{n!}{(n-3)!\times3!}.\frac{(2)\frac{n-3}{3}}{3}\) 

II. 4th term from the end =  \(\frac{n!}{(n-4)!\times4!}.(2)(3)\frac{3-n}{3}\)