To calculate the final amount in the account after 34 years (6 years of deposits and 28 years of growth without deposits), we divide the problem into two parts:

Step 1: Calculate the future value of the deposits after 6 years

The formula for the future value of monthly deposits in an account with monthly compounding is:

$FV = P \cdot \frac{(1 + r/n)^{nt} - 1}{r/n}$

Where:

• $P = 180$ (monthly deposit)
• $r = 0.07$ (annual interest rate as a decimal)
• $n = 12$ (compounding periods per year)
• $t = 6$ (years)

Substituting these values:

$FV = 180 \cdot \frac{(1 + 0.07/12)^{12 \cdot 6} - 1}{0.07/12}$

$FV = 180 \cdot \frac{(1 + 0.005833)^{72} - 1}{0.005833}$

$FV \approx 180 \cdot \frac{1.48985 - 1}{0.005833}$

$FV \approx 180 \cdot 84.007$

$FV \approx 15,121.30$

Step 2: Calculate the future value of the lump sum after 28 years

After 6 years, the balance of $15,121.30 will grow for another 28 years with no additional deposits. The formula for compound interest is:

$FV = PV \cdot (1 + r/n)^{nt}$

Where:

• $PV = 15,121.30$ (present value)
• $r = 0.07$
• $n = 12$
• $t = 28$

Substituting these values:

$FV = 15,121.30 \cdot (1 + 0.07/12)^{12 \cdot 28}$

$FV = 15,121.30 \cdot (1 + 0.005833)^{336}$

$FV \approx 15,121.30 \cdot 10.178$

$FV \approx 154,024.86$

Final Answer:

The total amount in the account after 34 years will be approximately $154,024.86.

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Would you like further clarification or more details? Here are some questions you might find helpful:

1. How does compounding frequency affect the total amount?
2. What if the interest rate changes during the investment period?
3. How does increasing the monthly contribution affect the final value?
4. Can you compare this to simple interest growth for the same period?
5. What is the impact of delaying the investment start by one year?

Tip: Regular, consistent contributions combined with compound interest can significantly boost long-term savings.