Mark Johnson saves a fixed percentage of his salary at the end of each year. This year he saved $2,000. For each of the next 5 years, he expects his salary to increase at an 10% annual rate, and he plans to increase his savings at the same 10% rate. There will be a total of 6 investments, the initial $2,000 plus five more. If the investments earn a return of 13% per year, how much will Mark have at the end of six years?

To solve this problem, we can use the future value formula for an investment compounded annually. Mark's investments can be considered as both a series of annuities (since he makes regular savings each year) and a lump sum investment (the initial $2,000).

For Mark's annually increasing savings, we first need to calculate the future value of each annual investment separately, since each will have a different amount of time to compound.

1. First investment ($2,000) will compound for 6 years. 2. Second investment, which is 10% more than the first, i.e., $2,000 * 1.10 = $2,200, will compound for 5 years. 3. Third investment will be $2,200 * 1.10 = $2,420 and will compound for 4 years. And so on...

The formula to calculate the future value of an investment compounded annually is:

Future Value = Present Value × (1 + r)^n

where: r = annual interest rate (as a decimal) n = number of years the money is invested

Let's calculate the future value for each investment:

1. For the first investment of $2,000 for 6 years: FV1 = $2,000 × (1 + 0.13)^6

2. For the second investment of $2,200 for 5 years: FV2 = $2,200 × (1 + 0.13)^5

3. Continuing this process for all investments up to the sixth year, we add each future value together to get the total.

Finally, sum all the future values: Total Future Value = FV1 + FV2 + FV3 + FV4 + FV5 + FV6

This gives us the total amount Mark will have at the end of six years.

However, instead of doing manual calculations for each investment, we can use a financial calculator or a spreadsheet to calculate the future values and add them up. Let's use the formula and do the math for each:

1. FV for first year's savings: = 2000 * (1 + 0.13)^6 = 2000 * 2.12922 = $4,258.44

2. FV for second year's savings: = 2200 * (1 + 0.13)^5 = 2200 * 1.88261 = $4,141.74

3. FV for third year's savings: = 2420 * (1 + 0.13)^4 = 2420 * 1.66529 = $4,029.80

4. FV for fourth year's savings: = 2662 * (1 + 0.13)^3 = 2662 * 1.47614 = $3,931.73

5. FV for fifth year's savings: = 2928.2 * (1 + 0.13)^2 = 2928.2 * 1.2769 = $3,739.44

6. FV for sixth year's savings: = 3221.02 * (1 + 0.13)^1 = 3221.02 * 1.13 = $3,639.75

Adding up all the future values gives us Mark's total savings at the end of six years.

Total = $4,258.44 + $4,141.74 + $4,029.80 + $3,931.73 + $3,739.44 + $3,639.75 Total = $23,750.90

Therefore, Mark will have $23,750.90 at the end of six years.

This problem introduces several financial concepts useful for understanding savings and investments.

1. **Compound Interest** - When an investment earns interest on both the initial principal and the accumulated interest from previous periods. In this problem, the 13% return Mark expects is compounded annually, meaning each year's interest is added to the principal for calculating the next year's interest.