increase annuity ?

[mikemark12]

Question:
Use a formula to determine the value of your car fund on your 18th birthday. List the numerical value used for each variable in the formula used.

What I know:
At birth parents contribute $20.00 per month to an increasing annuity that pays 61/4% annual interest, compounding monthly to the fund. Regular contributions are made until 18th birthday.

The first twelve months I calculated a balance on 246.89

 

[mikemark12]

I first had to construct an increasing annuity table for the first twelve months of the annuity's life. I used the formula below:

Interest=principle x rate x time
I=(P)(.0625)(1/12)

So for 12 months I came up with a balance of $246.89

The next part of the question asks for the formula to determine the value of the fund on the 18th birthday. This is what I can not figure out.

 

[tkhunny]

No. You have used Simple Interest. You need Monthly Compounding.

In this case, it is MUCH easier to start from the back.

We have:

i = 0.0625
j = 0.0625/12 = 0.00520833
r = (1+j) = 1.00520833
P = 20.00

Now, you have to make two choices.

1) When is the LAST payment? I'm going to assume a payment is made on your 18th birthday.

2) When is the FIRST payment? I'm going to assume the first was made the day you were born.

These two together might mean one payment more than you expect. Be very careful with assumptions. Disclose them well.

We are ready just to write down an expression for the accumulated value.

The last payment is P
The second to last payment is Pr
The third to last payment is Pr^2
etc.
The i-th payment from the end is Pr^(i-1)
etc.
This makes the first payment, #217 from the end: Pr^(217-1)

Adding all these up gives: P + Pr + Pr^2 + ... + Pr^(216) = P(1 + r + r^2 + ... + r^(216))

This is easily summed to \(\displaystyle P\cdot\frac{1-r^{217}}{1-r}\)

Really, you must get smooth and fluent with this sort of construction.