Question

Six years from today you need $10,000. You plan to deposit $1,500 annually, with the first payment to be made a year from today, in an account that pays a 5% effective annual rate. Your last deposit, which

will occur at the end of Year 6, will be for less than $1,500 if less is needed to reach $10,000. How large will your last payment be? Do not round intermediate calculations. Round your answer to the nearest
cent.

Answer 1

Explanation:

We can use the formula for the future value of an annuity to find the amount you'll have in the account at the end of Year 6. An annuity is a series of equal payments made at equal intervals of time, and the future value of an annuity is the amount you'll have in the future if you invest those payments and earn a given rate of interest.

The formula for the future value of an annuity is:

FV = PMT * (((1 + r)^n - 1) / r)

where:

FV = future value of the annuity

PMT = annual payment amount

r = annual interest rate as a decimal

n = number of payments

In this case:

PMT = $1,500

r = 0.05

n = 6

Plugging in the values:

FV = $1,500 * (((1 + 0.05)^6 - 1) / 0.05)

FV = $1,500 * (1.328867 - 1) / 0.05

FV = $1,500 * 0.328867 / 0.05

FV = $1,500 * 6.577340

FV = $9,865.61

So, the amount in the account at the end of Year 6 will be $9,865.61. To find the last payment, we subtract this amount from the desired future value of $10,000:

$10,000 - $9,865.61 = $134.39

Therefore, your last payment will be $134.39.