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]
Find the mean and variance for the following frequency distribution.
Classes | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
Frequencies | 5 | 8 | 15 | 16 | 6 |