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.
If a and b are distinct integers, prove that a - b is a factor of \(a^n - b^n\) , whenever n is a positive integer.
[Hint: write\( a ^n = (a - b + b)^n\) and expand]
take on sth: | to begin to have a particular quality or appearance; to assume sth |
take sb on: | to employ sb; to engage sb to accept sb as one’s opponent in a game, contest or conflict |
take sb/sth on: | to decide to do sth; to allow sth/sb to enter e.g. a bus, plane or ship; to take sth/sb on board |