Question

A person has $2$ parents, $4$ grandparents, $8$ great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

Answer 1

The number of ancestors are
$2,4,8,16,...$
Clearly, It is a $G.P$.
Here, first term $=a=2$,
common ratio $=r=2>1$ and
number of generations $=n=10$

We know, sum of $n$ terms in $\text{G.P}.=S_n=\frac{a(r^n-1)}{r-1}$
$S_{10}=\frac{2(2^{10}-1)}{2-1}$
$\Rightarrow S_{10}=2^{11}-2$
$\Rightarrow S_{10}=2048-2$
$\therefore S_{10}=2046$

Hence, number of ancestors preceding the person is $2046$.