Question

A student is constructing a family tree. If he wants to track back through $10$ generations to calculate the total number of ancestors he has, then the total number of ancestors after the ${10}^{th}$ generation is

• A. $2046$
• B. $1024$
• C. $2048$
• D. $1023$

Answer 1

The correct option is A $2046$


In first generation he has $2$ parents,
In $2^{nd}$ generation back, he has $4$ ancestors.
Similarly, in $3^{rd}$ generation back, he has $8$ ancestors.
This continues and forms a G.P., so number of ancestors is
$S=2+4+8+\dots +2^{10}\Rightarrow S=\frac{2(2^{10}-1)}{2-1}\therefore S=2\times 1023=2046$