Question

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

Answer 1

Hint: It forms a series in G.P. Find the first term, common ratio, and number of terms. Substitute all these values in the formula of sum of n terms and find the value of \[{{S}_{10}}\].

Complete step-by-step answer:
It is said that a person has 2 parents, 4 grandparents, 8 great grandparents etc. So we can write this as a series.

Thus the number of ancestors can be written as 2,4,8,16……

This forms a G.P. which is a geometric progression which is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. It is denoted by r.

\[\therefore \]Common ratio, \[r=\dfrac{Term\text{ }2}{Term\text{ }1}=\dfrac{4}{2}=2\]

\[r=\dfrac{Term\text{ 3}}{Term\text{ 2}}=\dfrac{8}{4}=2\]

Thus we got the common ratio, r = 2 and first term, a = 2.

It is said that we need to find the number of ancestors during 10 generations preceding his own, i.e. we need to find \[{{S}_{10}}\].

\[\therefore n=10\], which is the number of terms.

Sum of n terms of a G.P. is given by the formula,

\[{{S}_{n}}=\dfrac{a\left( {{r}^{n}}-1 \right)}{r-1}\].

Put n = 10, a = 2, r = 2 and find \[{{S}_{10}}\].

\[\begin{align}

  & \therefore {{S}_{10}}=\dfrac{2\left( {{2}^{10}}-1 \right)}{2-1} \\

 & \therefore {{S}_{10}}=2\left( {{2}^{10}}-1 \right) \\

 & ={{2.2}^{10}}-2={{2}^{10+1}}-2 \\

 & ={{2}^{11}}-2. \\

 & \therefore {{S}_{10}}=2048-2=2046. \\

\end{align}\]

\[\therefore \]The value of \[{{2}^{11}}=2048\], which is 2 multiplied 11 times.

Thus we got the number of ancestors preceding the person as 2046.


Note: By reading the question and trying to get the answer directly, it may be a little complex. Thus it is simpler to find the series and apply the formula to get the sum of n terms.