Suppose an account pays 2.5% interest that is compounded annually. At the beginning of each year, $10,000 is deposited into the account (starting with $10,000 for the first year).

Assuming no withdrawals or other deposits are made and that the interest rate is fixed, the balance of the account (rounded to the nearest dollar) after the seventh deposit is __________.

To determine the balance of the account after the seventh deposit with 2.5% interest compounded annually, we must consider the following:

1. **Formula for Compound Interest with Recurring Deposits:**

   Each deposit earns interest over a different period depending on when it was deposited. The general formula for compound interest is:

   \[
   A = P \left( 1 + \frac{r}{n} \right)^{nt}
   \]
   where:
   - $A$ is the future value of the deposit.
   - $P$ is the principal (initial deposit).
   - $r$ is the annual interest rate (2.5% or 0.025 in decimal).
   - $n$ is the number of times the interest is compounded per year (since it's annually, $n = 1$).
   - $t$ is the number of years the deposit remains in the account.

   In this problem, deposits are made annually, and the interest is compounded annually. Each deposit earns a different amount of interest based on how long it has been in the account.

2. **Breakdown of Deposits:**

   Let's break down the future value of each deposit by year:
   - The first deposit earns interest for 7 years.
   - The second deposit earns interest for 6 years.
   - The third deposit earns interest for 5 years.
   - The fourth deposit earns interest for 4 years.
   - The fifth deposit earns interest for 3 years.
   - The sixth deposit earns interest for 2 years.
   - The seventh deposit earns interest for 1 year.

3. **Balance after Each Deposit:**

   For each deposit of $10,000, we apply the formula and sum up the future values:

   \[
   \text{Total Balance} = 10,000(1.025)^7 + 10,000(1.025)^6 + 10,000(1.025)^5 + 10,000(1.025)^4 + 10,000(1.025)^3 + 10,000(1.025)^2 + 10,000(1.025)^1
   \]

Let’s compute this.The balance of the account after the seventh deposit, compounded annually at a 2.5% interest rate, is **$77,361** (rounded to the nearest dollar).

Would you like more details on the calculation, or do you have any other questions?

Here are 5 related questions to further explore this concept:
1. How would the balance change if the interest were compounded semi-annually instead of annually?
2. What if the interest rate increased to 3%? How would the total balance be affected?
3. How would the total balance change if deposits were made at the end of each year instead of the beginning?
4. How much interest was earned specifically from the first deposit over the 7 years?
5. How much would the account balance be after 10 years of the same deposit schedule?

**Tip:** The more frequently interest is compounded, the greater the total balance will be, even with the same interest rate!

Suppose an account pays 2.5% interest that is compounded annually. At the beginning of each year, $10,000 is deposited into the account (starting with $10,000 for the first year). Assuming no withdrawals or other deposits are made and that the interest rate is fixed, the balance of the account (rounded to the nearest dollar) after the seventh deposit is __________.

Explore how to calculate the balance of an account after seven annual deposits of $10,000 with 2.5% interest compounded annually. Learn the formula for compound interest and see the detailed breakdown of each deposit's future value. Suitable for Grades 10-12 students studying exponential growth and financial math.

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