What is the constant term in the binomial exp | vedclass

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Question

(A) Given expression: $(1+3x)^n \left(1+\frac{1}{3x}\right)^n$ $= (1+3x)^n \left(\frac{3x+1}{3x}\right)^n$ $= \frac{(1+3x)^n (1+3x)^n}{(3x)^n}$ $= \fr

Vedclass · Accepted Answer

$\binom{2n}{n}$

Vedclass · Answer

(A) Given expression: $(1+3x)^n \left(1+\frac{1}{3x}ight)^n$ $= (1+3x)^n \left(\frac{3x+1}{3x}ight)^n$ $= \frac{(1+3x)^n (1+3x)^n}{(3x)^n}$ $= \frac{(1+3x)^{2n}}{(3x)^n}$ The general term in the expansion of $(1+3x)^{2n}$ is $T_{r+1} = \binom{2n}{r} (3x)^r$. To find the constant term,we need the term where the power of $x$ is $0$. The expression is $\frac{1}{(3x)^n} 	imes \sum_{r=0}^{2n} \binom{2n}{r} (3x)^r = \sum_{r=0}^{2n} \binom{2n}{r} \frac{(3x)^r}{(3x)^n} = \sum_{r=0}^{2n} \binom{2n}{r} (3x)^{r-n}$. The constant term occurs when $r-n = 0$,i.e.,$r = n$. Substituting $r=n$,the constant term is $\binom{2n}{n} (3x)^{n-n} = \binom{2n}{n}$. Thus,the constant term is $\binom{2n}{n}$.

What is the constant term in the binomial expansion of $(1+3x)^n \left(1+\frac{1}{3x}\right)^n$?

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