I may be using a different formula from the one you were taught but it will amount to the same, especially in the number of years. Suppose we look first at the case of making an annual deposit of $1000 starting at age 20 and making the last deposit at age 64. The $1000 deposit made at age 64 will have drawn interest for 1 year, and will be worth 1000(1.06)^1 The $1000 deposit made at age 63 will have drawn interest for 2 years, and will be worth 1000(1.06)^2 The $1000 deposit made at age 62 will have drawn interest for 3 years, and will be worth 1000(1.06)^3 ... (We observe that the number of years the deposit has drawn interest is always 65 minus the age) ... The $1000 deposit made at age 22 will have drawn interest for 43 years, and will be worth 1000(1.06)^43 The $1000 deposit made at age 21 will have drawn interest for 44 years, and will be worth 1000(1.06)^44 The $1000 deposit made at age 20 will have drawn interest for 42 years, and will be worth 1000(1.06)^45 So we have the geometric series: 1000(1.06) + 1000(1.06)^2 + ... + 1000(1.06)^45 = 1000(1.06 + 1.06^2 + 1.06^3 + ... + 1.06^45)So the series in the parentheses is: Therefore 1000 times that is $225508.12, which is the amount at age 65. ----------- Next we look first at the case of making an annual deposit of $3000 starting at age 50 and making the last deposit at age 64. The $3000 deposit made at age 64 will have drawn interest for 1 year, and will be worth 3000(1.06)^1 The $3000 deposit made at age 63 will have drawn interest for 2 years, and will be worth 3000(1.06)^2 The $3000 deposit made at age 62 will have drawn interest for 3 years, and will be worth 3000(1.06)^3 ... (We observe again that the number of years is always 65 minus the age) ... The $3000 deposit made at age 52 will have drawn interest for 43 years, and will be worth 3000(1.06)^13 The $3000 deposit made at age 51 will have drawn interest for 44 years, and will be worth 3000(1.06)^14 The $3000 deposit made at age 50 will have drawn interest for 42 years, and will be worth 3000(1.06)^15 So we have the geometric series: 3000(1.06) + 3000(1.06)^2 + ... + 3000(1.06)^15 = 3000(1.06 + 1.06^2 + 1.06^3 + ... + 1.06^15) So the series in the parentheses is: Therefore 3000 times that is $74017.58, which is the amount at age 65. Edwin