Question:
Prove that \(\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n\)
Prove that \(\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n\)
Updated On: Oct 25, 2023
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Solution and Explanation
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.
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