Hello, Sparta!

Are you looking for aderivationfor the required formula?

A man invests $60,000 in an account which earns interest at a compound rate fo 7% per annum.

He wants to make a withdrawal of $W at the end of each year for his annual holiday expenses.

The withdrawal is made immediately after that year's interest has been paid.

He intends to do this for 20 years, so that at the end of this time his account will be $0.

He is trying to work out the amount of $W that he will have towards his holiday each year.

Your reasoning is correct. .Let = amount invested.

At the end of one year, the account has: dollars.

He withdraws $W and the account has: dollars.

At the end of year 2, the account has: dollars.

He withdraws $W and the account has: dollars.

At the end of year 3, the account has: dollars.

He withdraws $W and the account has: dollars.

. . . and so on . . .

At the end of year 20, the account has:

. . dollars.

He withdraws $W and the account has:

. . dollars.

But this final balance will be zero dollars.

We have: .

The expression at the far right is ageometric series

. . with first term , common ratio , and 20 terms.

So we have: .

. . Therefore: .

For

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In general, the formula is: .

. . where: .

This formula is identical to the Amortization Formula

. . in which we are to pay off a debt.