How do you calculate annuity payments with a 5% increment each year?

[NameIsUnique]

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 billion 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

 

[NameIsUnique]

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

 

[NameIsUnique]

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]

Nvm I get it.

Ok, very well!