# Growing annuity with fixed growth

I've found this formula for the Future Value of a Growing Annuity: see here

I'm looking for a formula that will provide the future value similar to this formula, only where I can provide a fixed amount of growth every period. For instance, the growth would be \$25 more every year rather than growing by 3% every year.

Anyone know the formula to accomplish this?

Preamble

The demonstration in the link states "The growth rate in this example would be the 5% increase per year", so although you mention substituting the growth rate of 3% I assume you mean that the 5% growth rate should be substituted by a fixed \$25 increase. You can swap the figures as you please in the formula. The point is I'm fairly sure you want the growth rate substituted, not the interest rate. (In the example in the link 3% is the interest rate.)

Solution

I doubt you'll find a formula quoted so here is one derived, based on the example.

The formula on the link, with the example shown, can be found like this.

First, adding up the contribution from each year:-

``````2000*(1 + 0.05)^0*(1 + 0.03)^4 +
2000*(1 + 0.05)^1*(1 + 0.03)^3 +
2000*(1 + 0.05)^2*(1 + 0.03)^2 +
2000*(1 + 0.05)^3*(1 + 0.03)^1 +
2000*(1 + 0.05)^4*(1 + 0.03)^0 = 11700.75
``````

This can be expressed as a sum like so:-

or simplified (using Mathematica ) to produce the formula:-

The calculation you require, with fixed \$25 growth, is this:-

``````(2000 + 25*0)*(1 + 0.03)^4 +
(2000 + 25*1)*(1 + 0.03)^3 +
(2000 + 25*2)*(1 + 0.03)^2 +
(2000 + 25*3)*(1 + 0.03)^1 +
(2000 + 25*4)*(1 + 0.03)^0 = 10875.88
``````

which can be expressed in this sum:-

This can be simplified in a couple of ways:-

So, taking the shorter version

``````FV = -((x - (1 + i)^n (i p + x) + i (p + n x))/i^2)
``````

e.g.

``````x = 25
p = 2000
i = 0.03
n = 5

FV = 10875.88
``````
• What you've assumed in the preamble is correct. Thank you, this looks like a great solution that will work for me. Oct 16, 2014 at 17:45
• Jolly good. :-) Oct 16, 2014 at 18:01