To determine Martha's savings after [tex]\( n \)[/tex] years given that the initial savings amount is \[tex]$50 and it increases by 5% annually, we need to use the formula for compound interest. Let's go through the steps to derive this formula:
1. Initial Amount (\$[/tex]50):
This is Martha's initial savings.
2. Interest Rate (5% annually):
An interest rate of 5% per year is equivalent to a multiplier of 1.05 (since 100% + 5% = 105%, which is expressed as a decimal: 1.05).
3. Compounding Period:
The savings grow annually, so we will consider [tex]\( n \)[/tex] years as the compounding period.
4. Compound Interest Formula:
The general formula for compound interest is:
[tex]\[
A = P (1 + r)^n
\][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( n \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial savings).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of years.
5. Substitute the given values:
- [tex]\( P = 50 \)[/tex]
- [tex]\( r = 0.05 \)[/tex] (which represents the 5% interest rate)
- Therefore, [tex]\( (1 + r) = 1 + 0.05 = 1.05 \)[/tex]
Substituting these into the formula gives us:
[tex]\[
a_n = 50 (1.05)^n
\][/tex]
Given the choices:
A. [tex]\( a_n = 50(1.05)^n \)[/tex]
B. [tex]\( a_n = 50(1.05)^{n+1} \)[/tex]
C. [tex]\( a_n = 50(0.05)^n \)[/tex]
D. [tex]\( a_n = 50(0.05)^{n+1} \)[/tex]
The correct formula is:
[tex]\[
a_n = 50(1.05)^n
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{a_n = 50(1.05)^n}
\][/tex]
