Question

 A person has 2 parents, 4 grand parents, 8 great grand parents and so on . Find the total number of ancestors during 8^th generations preceding his own ?
1. 8² - 2
2. 2⁸ - 2
3. 2⁹ - 2
4. 9²

Answer 1

Correct Answer - Option 3 : 2⁹ - 2

Concept:

Let us consider sequence a1, a2, a3 …. an is a G.P.

• Common ratio = r = \(\frac{{{{\rm{a}}_2}}}{{{{\rm{a}}_1}}} = \frac{{{{\rm{a}}_3}}}{{{{\rm{a}}_2}}} = \ldots = \frac{{{{\rm{a}}_{\rm{n}}}}}{{{{\rm{a}}_{{\rm{n}} - 1}}}}\)
• nth  term of the G.P. is an = arn−1
• Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\); where r >1
• Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {1 - {\rm{\;}}{{\rm{r}}^{\rm{n}}}} \right)}}{{1 - {\rm{\;r}}}}\); where r <1
• Sum of infinite GP = \({{\rm{s}}_\infty } = {\rm{\;}}\frac{{\rm{a}}}{{1{\rm{\;}} - {\rm{\;r}}}}{\rm{\;}}\) ; |r| < 1

 

Calculation:

The number of ancestors is given by 

2, 4, 8, 16,.......

It is a GP in which first term = a = 2 & common ratio = r = 4/2 = 2

Number of generations (n) = 8

We have to find S₈ .

S_n = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\) 

So, S₈ = \(\frac{2(2^{8} - 1)}{2 - 1}\) = 2⁹ -  2