Respuesta :
Answer:
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) = [tex]\frac{a_{2} }{a_1}[/tex]
= [tex]\frac{4}{2}[/tex]
= 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 [tex]n_{th}[/tex] term, we multiply [tex](n-1)_{th}[/tex] term by r.
i.e. aₙ = aₙ₋₁ * r
=> f(n) = aₙ₋₁ * r [Recursive formula]
