Answer
1. **Recurrence Relation:**
\[
R_{n+1} = 1.01\,R_n,\quad R_0 \text{ given.}
\]
2. **Amount after 1 Year** (4 quarters) if \(R_0= \$5000\):
\[
R_4 = 5000 \times 1.01^4 \approx \$5203.02.
\]
3. **APR for the Account** (nominal):
\[
\boxed{4\%\text{ (nominal APR)}},
\]
while the *effective* annual interest rate (APY) is approximately 4.06%.
Work Step by Step
## 1. Recurrence Relation
Let \(R_n\) be the amount on deposit at the *end* of the \(n\)th quarter (3‐month period). We are told:
- The annual interest rate is 4%.
- Interest is compounded quarterly (i.e., every quarter).
Since 4% per year over 4 quarters implies \(4\%\!/4 = 1\%\) interest each quarter, the amount grows each quarter by a factor of \(1 + 0.01 = 1.01\). Therefore, the recurrence is
\[
\boxed{R_{n+1} \;=\; 1.01\,R_{n},}
\]
with the initial condition \(R_0\) being the amount originally deposited.
---
## 2. Amount After One Year (4 Quarters)
If \(R_0 = 5000\), then after one quarter the balance is \(1.01 \times 5000\), after two quarters it is \(1.01^2 \times 5000\), and so on. After 4 quarters (which is one year), the amount is
\[
R_4 \;=\; 1.01^4 \times R_0
\;=\; 1.01^4 \times 5000.
\]
Numerically,
\[
1.01^4 \approx 1.04060401,
\]
so
\[
R_4 \approx 5000 \times 1.04060401
\;=\; 5203.02.
\]
Hence, at the end of one year, there is about \(\boxed{\$5203.02}\) on deposit (if we round to the nearest cent).
---
## 3. APR (Annual Percentage Rate)
- **Nominal APR:** By convention, when a bank advertises “4% annual interest, compounded quarterly,” the nominal APR is **4%**. 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, you note that each quarter the account grows by 1%. Over one year (4 quarters), the multiplication factor is \(1.01^4 \approx 1.040604\). Subtracting 1 gives about 0.040604, i.e., **4.0604%**. This is the effective annual interest rate or “APY.”
