Lisa Richter deposited $5,000 at 4% compounded semiannually for three years. At the beginning of the fourth year, Lisa deposited $2,500. What would her balance be at the end of five years assuming she is still earning 4% compounded semiannually?

8801.99

To calculate Lisa's balance at the end of five years, we need to consider the two deposits and the interest earned.

Let's start with the first deposit of $5,000 at a 4% interest rate compounded semiannually for three years. The formula to calculate the future value of an investment with compound interest is:

A = P(1 + r/n)^(nt)

Where:
A = The future value of the investment
P = The principal amount (initial deposit)
r = The annual interest rate (expressed as a decimal)
n = The number of times that interest is compounded per year
t = The number of years

In this case, P = $5,000, r = 4% or 0.04 (as a decimal), n = 2 (semiannual), and t = 3 years. Plugging these values into the formula, we get:

A1 = $5,000 * (1 + 0.04/2)^(2*3)
A1 ≈ $5,000 * (1.02)^6
A1 ≈ $5,000 * 1.1236
A1 ≈ $5,618

So, at the end of three years, Lisa's account balance from the first deposit would be approximately $5,618.

For the second deposit of $2,500 at the beginning of the fourth year, we assume no interest has been earned from the previous balance. Therefore, this deposit will start growing from this point. We can simply add the amount to the previous balance of $5,618:

A2 = $5,618 + $2,500
A2 ≈ $8,118

At the beginning of the fourth year, Lisa would have a balance of approximately $8,118.

Now, let's calculate the balance at the end of five years using the formula mentioned earlier. We will take into account the $8,118 as the new principal amount and calculate the interest earned for the remaining two years:

A3 = $8,118 * (1 + 0.04/2)^(2*2)
A3 ≈ $8,118 * (1.02)^4
A3 ≈ $8,118 * 1.08243264
A3 ≈ $8,791.41

Therefore, at the end of the fifth year, Lisa's account balance would be approximately $8,791.41.

Note that there may be small variations due to rounding the decimals in the calculations.

To calculate the balance at the end of five years, we need to calculate the balance for the first three years and then add the additional deposits and interest for the last two years.

First, let's calculate the balance at the end of the third year. The formula for compound interest is given by:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years

In this case, the principal (initial deposit) is $5,000, the interest rate is 4% (or 0.04 as a decimal), and interest is compounded semiannually (so n = 2) for a total of three years (t = 3).

A = 5000(1 + 0.04/2)^(2*3)
= 5000(1 + 0.02)^6
= 5000(1.02)^6
≈ 5614.43

So, the balance at the end of the third year would be approximately $5,614.43.

Now, let's calculate the balance at the end of the fifth year. We will add the amount from the end of the third year ($5,614.43) to the additional deposit of $2,500 and then calculate the interest for the remaining two years.

The new principal amount will be $5,614.43 + $2,500 = $8,114.43.

Using the same formula, we can calculate the balance at the end of the fifth year:

A = 8114.43(1 + 0.04/2)^(2*2)
= 8114.43(1.02)^4
≈ 8,911.36

Therefore, the balance at the end of five years would be approximately $8,911.36.