Question

Prove that $\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n$

Answer 1

By Binomial Theorem,

$\underset{r=0} {\overset{n}∑} \,{ ^nC_r} a^{n-r} b^r= (a+b)^n$

By putting $b=3$ and $a = 1$in the above eqution, we obtain

$\underset{r=0} {\overset{n}∑} \,{ ^nC_r} (1)^{n-r} (3)^r= (1+3)^n$

⇒ $\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n$

Hence, proved.