Answer
1. **Recurrence Relation:**
\[
S_{n+1} = 1.0025\,S_n,\quad S_0 \text{ given.}
\]
2. **Amount after 1 Year** (12 months) if \(S_0= \$10{,}000\):
\[
S_{12} = 10{,}000 \times 1.0025^{12} \approx \$10{,}304.15.
\]
3. **APR for the Account** (nominal):
\[
\boxed{3\%\text{ (nominal APR)}},
\]
while the *effective* annual interest rate (APY) is approximately 3.0415%.
Work Step by Step
## 1. Recurrence Relation
Let \(S_n\) be the amount on deposit at the *end* of the \(n\)th month. We are told:
- The annual interest rate is 3%.
- Interest is *compounded monthly*.
Since there are 12 months in a year, each month the balance grows by a factor of
\[
1 + \frac{0.03}{12} = 1.0025.
\]
Hence the recurrence relation is
\[
\boxed{S_{n+1} = 1.0025 \, S_{n},}
\]
with the initial condition \(S_0\) being the amount originally deposited.
---
## 2. Amount After One Year (12 Months)
If \(S_0 = \$10{,}000\), then after 12 months (one year), the amount on deposit is
\[
S_{12}
\;=\;
S_0 \times 1.0025^{12}
\;=\;
10{,}000 \times 1.0025^{12}.
\]
Numerically,
\[
1.0025^{12} \approx 1.030415,
\]
so
\[
S_{12}
\approx
10{,}000 \times 1.030415
\;=\;
\$10{,}304.15.
\]
Hence, at the end of one year, there is about \(\boxed{\$10{,}304.15}\) on deposit (if we round to the nearest cent).
---
## 3. APR (Annual Percentage Rate)
- **Nominal APR:** By convention, when a bank advertises “3% annual interest, compounded monthly,” the nominal APR is **3%**. This is the stated annual rate before considering the compounding effect.
- **Effective Annual Rate (sometimes called APY):** If you wish to find the *actual* or *effective* annual yield, note that each month the account grows by \(0.25\%\). Over 12 months, the multiplication factor is \(1.0025^{12} \approx 1.030415\). Subtracting 1 gives about \(0.030415\), i.e., **3.0415%**. This is the effective annual interest rate or “APY.”
