solve this problem using these instructions   At age 25, you begin depositing $175 at the end of each month into an account that pays 2.7% APR compounded monthly. How much money will you have by age 65? and suppose you deposit 25 extra dollars per month but the extra amount is not matched with your employer how much money will you have at 65, and at 70? with this given formula K((2+r)^n-1/r)

We are tasked with calculating the future value of a series of monthly deposits into an account with a 2.7% APR compounded monthly. We are also asked to assess the impact of adding an extra $25 to each deposit.

### Step 1: Given Information
- **Regular deposit amount (K)**: $175
- **Extra deposit (second scenario)**: $25, so total monthly deposit = $175 + $25 = $200
- **Interest rate (APR)**: 2.7% per year, compounded monthly.
- **Monthly interest rate (r)**: 2.7% ÷ 12 = 0.225% = 0.00225 (as a decimal).
- **Time period for first scenario** (age 25 to age 65): 40 years = 40 × 12 = 480 months.
- **Time period for second scenario** (age 25 to age 70): 45 years = 45 × 12 = 540 months.

### Step 2: Future Value Formula
The future value of an annuity (series of monthly deposits) with compounding interest can be calculated using the formula:

\[
FV = K \times \left(\frac{(1 + r)^n - 1}{r}\right)
\]

Where:
- $ K $ = monthly deposit amount,
- $ r $ = monthly interest rate,
- $ n $ = total number of payments (months),
- $ FV $ = future value of the account at the end of the period.

### Scenario 1: Regular Monthly Deposits of $175

#### For age 65 (40 years, 480 months):

We substitute the values:
- $ K = 175 $
- $ r = 0.00225 $
- $ n = 480 $

\[
FV_{65} = 175 \times \left(\frac{(1 + 0.00225)^{480} - 1}{0.00225}\right)
\]

### Scenario 2: Extra Monthly Deposits of $200

#### For age 65 (40 years, 480 months):

We substitute the values:
- $ K = 200 $
- $ r = 0.00225 $
- $ n = 480 $

\[
FV_{65} = 200 \times \left(\frac{(1 + 0.00225)^{480} - 1}{0.00225}\right)
\]

#### For age 70 (45 years, 540 months):

We substitute the values:
- $ K = 200 $
- $ r = 0.00225 $
- $ n = 540 $

\[
FV_{70} = 200 \times \left(\frac{(1 + 0.00225)^{540} - 1}{0.00225}\right)
\]

Let me now calculate the results.Here are the results based on the calculations:

1. **At age 65** with regular monthly deposits of $175, you will have approximately **$150,975**.
2. **At age 65** with extra monthly deposits of $200, you will have approximately **$172,543**.
3. **At age 70** with extra monthly deposits of $200, you will have approximately **$210,284**.

Would you like more details or have any questions?

### Related Questions:
1. What would the value be if you increased the APR to 3%?
2. How much would you accumulate if you started at age 30 instead of 25?
3. What would happen if the interest was compounded weekly instead of monthly?
4. What is the total interest earned in both scenarios by age 65?
5. How does increasing the monthly deposit to $300 change the results?

#### Tip:
Small increases in monthly contributions early on can significantly boost your savings due to the effects of compound interest over time.

At age 25, you begin depositing $175 at the end of each month into an account that pays 2.7% APR compounded monthly. How much money will you have by age 65? Suppose you deposit 25 extra dollars per month, but the extra amount is not matched with your employer. How much money will you have at 65 and at 70? Use the formula K((2+r)^n-1)/r.

Calculate how much money you will have by age 65 and 70 by making monthly deposits of $175 into an account with a 2.7% APR compounded monthly. Learn how adding $25 extra per month can impact your savings, using the future value of an annuity formula.

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