Let's break down both parts systematically:

Part 1 (a): Verify the value of the account at the beginning of the third year.

At the beginning of the first year, you deposit $100, and that $100 grows by $r$ each year. By the beginning of the third year, the value of the account is the sum of the compounded value of each deposit.

Step-by-step breakdown:

1. First deposit (made at the beginning of year 1):

   • This deposit grows for 2 years.
   • After 2 years, its value is: $100(1 + r)^2 = 100(1 + 2r + r^2) = 100r^2 + 200r + 100$

2. Second deposit (made at the beginning of year 2):

   • This deposit grows for 1 year.
   • After 1 year, its value is: $100(1 + r) = 100r + 100$

3. Third deposit (made at the beginning of year 3):

   • This deposit does not grow because it is made at the start of the third year, so its value remains: $100$

Now, add all these values together to get the total at the beginning of the third year: $100r^2 + 200r + 100 \quad \text{(from 1st deposit)}$ $+ 100r + 100 \quad \text{(from 2nd deposit)}$ $+ 100 \quad \text{(from 3rd deposit)}$ Simplify: $100r^2 + (200r + 100r) + (100 + 100 + 100) = 100r^2 + 300r + 300$

Thus, the value of the account at the beginning of the third year is: $T(r) = 100r^2 + 300r + 300$

This verifies the equation provided.

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Part 2: Choose the correct answer.

We need to identify which option matches the calculation done above. The correct equation is:

$100(1 + r)^2 + 100(1 + r) + 100$

Let's evaluate the expressions in the given choices:

• Option A: $100(1 + r)(1 + r) + 100(1 + r) + 100$ This simplifies to: $100(1 + r)^2 + 100(1 + r) + 100 = 100r^2 + 200r + 100 + 100r + 100 = 100r^2 + 300r + 300$ This matches our expression. Hence, Option A is the correct answer.

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Would you like further details or clarifications on any step? Let me know!

Related Questions:

1. What if the interest rate is compounded monthly instead of yearly?
2. How would the formula change if the deposit amounts were different each year?
3. Can this method be generalized to $n$ years? What would the total amount be after $n$ years?
4. What happens to the total amount if $r = 0$? Does the formula still work?
5. How does the account value change if deposits are made at the end of each year instead?

Tip:

Always ensure that when handling compound interest problems, the growth periods for each deposit are correctly tracked. The key difference is how long each deposit has been earning interest.