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.