Question

Caden's savings account had
\[\$70\] in its first year. Each year since then, his account accumulated interest amounting to
\[10\%\] of the balance in the previous year.
Let
\[f(n)\] be Caden's account balance at the
\[n^\text{th}\] year of the saving.
\[f\] is a sequence. What kind of sequence is it?
Choose 1 answer:
Choose 1 answer:
(Choice A) Arithmetic sequence
A
Arithmetic sequence
(Choice B) Geometric sequence
B
Geometric sequence
Complete the recursive formula for
\[f(n)\].
\[f(1)=\]
\[f(n)=f(n-1)\]

Answer 1

Caden's savings account accumulates interest at a constant rate of \(10\%\) each year on the previous balance. This means that the amount in the account is multiplied by a fixed factor each year, specifically \(1 + 0.10 = 1.10\).

Given this characteristic, \(f(n)\) represents the balance after \(n\) years, which indicates that it is a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant (in this case, \(1.10\)).

We can express the recursive formula for \(f(n)\):

1. Starting value:
\[
f(1) = 70
\]

2. Recurrence relation for subsequent years:
\[
f(n) = f(n-1) \times 1.10
\]

So, putting this together:

- Complete the statement:
\[
f(1) = 70
\]
\[
f(n) = f(n-1) \times 1.10
\]

Thus, the answer is:
Choice B: Geometric sequence.