Finance Word Problem

[amandacravens]

This word problem is giving me a run for my money! I've attempted 8 times (literally), and I cannot get the answer correct. This is the last problem for my homework. Please help!

Problem:

You are saving for retirement. To live comfortably, you decide you will need to save $2,000,000 by the time you are 65. Today is your 25^th birthday, and you decide, starting today and continuing on every birthday up to and including your 65^th birthday, that you will put the same amount into a savings account. If the interest rate is 8%, how much must you set aside each year to make sure that you will have $2,000,000 in the account on your 65^th birthday?

The amount to deposit each year is $______. (Round to the nearest dollar)

 

[tkhunny]

Have you considered just building it?

It often helps to start from the back.

Let's just talk about payments, first.

Age 65: P
Age 64: P
Age 63: P
...
Age 26: P
Age 25: P

Now add the interest accumulation. i = 0.08. Define r = 1+i = 1.08

Age 65: P*r^0 - No interest on the very last payment.
Age 64: P*r^1
Age 63: P*r^2
...
Age 26: P*r^39
Age 25: P*r^40

This gives the equation to be solved:

2000000 = P*r^0 + P*r^1 + P*r^2 + ... + P*r^39 + P*r^40

A little algebra.

2000000 = P*(r^0 + r^1 + r^2 + ... + r^39 + r^40)
2000000 = P*(1 + r^1 + r^2 + ... + r^39 + r^40)

We're almost done. Can you add up that finite geometric series in the parentheses?

 

[amandacravens]

Oh goodness...I appreciate the different approach...that just confused me, though! I'm not even familiar with what a finite geometric series is. :-/

Have you considered just building it?

It often helps to start from the back.

Let's just talk about payments, first.

Age 65: P
Age 64: P
Age 63: P
...
Age 26: P
Age 25: P

Now add the interest accumulation. i = 0.08. Define r = 1+i = 1.08

Age 65: P*r^0 - No interest on the very last payment.
Age 64: P*r^1
Age 63: P*r^2
...
Age 26: P*r^39
Age 25: P*r^40

This gives the equation to be solved:

2000000 = P*r^0 + P*r^1 + P*r^2 + ... + P*r^39 + P*r^40

A little algebra.

2000000 = P*(r^0 + r^1 + r^2 + ... + r^39 + r^40)
2000000 = P*(1 + r^1 + r^2 + ... + r^39 + r^40)

We're almost done. Can you add up that finite geometric series in the parentheses?