# Formula for annuity payments

So, unless you've been living under a rock, you should know the jackpot for the powerball is at 1.5 billion dollars.

I was looking up the distribution of annuity payments and the website said that the payments are not equally distributed. They are incremented by 5% each year.

Like the nerd I am, I tried figuring out the math but didn't know where to start.

I know that 1.5 billion / 30 payments = 50 million a year (before taxes)

How would you go about figuring out 30 payments equating to 1.5 billion but the condition is that each payment is 5% more than the next?

S.G. Janssens
They are incremented by 5% each year.
the condition is that each payment is 5% more than the next?
Is the amount increasing or decreasing in time?

If the total amount is ##p## (= 1.5 billon dollars) and the rate of increase / decrease is ##\lambda## (= 1.05 or 0.95) and your amount in the ##k##th year is ##a_k##, then ##a_k = \lambda^{k-1}a_1## for ##k = 1,\ldots,n## where ##n## is the amount of years. Now set ##\sum_{k=1}^n{a_k} = p## (geometric sum) and solve for ##a_1##.

• NameIsUnique
Increasing.

Year 1 = x
Year 2 = x +(x* 0.05)
and keeps incrementing until 1.5 billion

S.G. Janssens
Ok, so then you set ##\lambda = 1.05##, take the geometric sum and solve the equation for ##a_1##. Once ##a_1## is known, use the formula for ##a_k## to compute the amount in year ##k##.

• NameIsUnique
Thanks a lot!

S.G. Janssens
Don't mention it. Just be sure to let me know if you win the jackpot Don't mention it. Just be sure to let me know if you win the jackpot I just plugged it in year one

I think I'm doing it wrong.

1.5 billion = (1.05) ^ (1-1) * a1

and I solve for a1?

It seems like I get 1.5 billion

Nvm I get it.

S.G. Janssens
No, that is not correct. You need to solve
$$a_1\sum_{k=1}^n{\lambda^{k-1}} = p$$
First you need to evaluate the sum, using the standard formula for the geometric sum. I leave that up to you as a challenge. Once that is done, you can solve for ##a_1##.

S.G. Janssens