Nathan has two parents, four grandparents, and so on. explain how you can write both an explicit formula and a recursive formula that represents the number of ancestors nathan has if we go back n generations.

Explicit formula

f(n) = 2*2⁽ⁿ⁻¹⁾

Recursive formula                  

f(n) = aₙ₋₁ * r

Step-by-step explanation:

Since, Nathan has two parents, four grandparents, and so on, it would form a Geometric Progression (GP) series for the number of Nathan's ancestors as below:-

2, 4, 8, 16, 32, ....

First term (a₁) = 2

Common Ratio (r) =

                             =

                             = 2

So,

Number of ancestors of Nathan if we go back n generations = a₁r⁽ⁿ⁻¹⁾

Plugging in the values of a₁ and r, we get

f(n) = 2*2⁽ⁿ⁻¹⁾                       [Explicit formula]

Now,

A geometric progression (GP) series is of the form,

a, ar, ar², ar³, ar⁴, ar⁵........  

in which the first term a₁=a and other the terms are obtained by multiplying with r.

Observe that each term is r times the previous term. So, to get the term, we multiply term by r.

i.e. aₙ = aₙ₋₁ * r    

=> f(n) = aₙ₋₁ * r                     [Recursive formula]