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Mathematically, this problem can be broken down into three parts:

  1. A 20-year annuity at $100/month. Find the future value and the total interest.
  2. The future value from part 1 represents the present value of a single investment that earns interest for another 20 years. Find the future value and the total interest.
  3. A 20-year annuity at $200/month. Find the future value and the total interest.

The total amount in the account after 40 years is the future value from step 2 plus the future value from step 3. We don't include the future value from step 1 because that amount is already included in the single investment problem in part 2.

The total interest earned is the sum of the interest earned in parts 1, 2, and 3. The total interest earned is also the difference between the total amount in the account after 40 years (what they will get out of their retirement account) and the sum of all of the deposits into the account:

I = A - (M1n1 + M2n2)

M1 represents the size of the payments at the beginning and n1 is the number of these deposits that were made. M2 is the size of the payments made in the second part of this retirement plan and n2 is the number of these payments that were made. In this particular case, M1  = 100, n1 = 240, M2 = 200, and n2 = 240.