You are seeking the "future value of a growing annuity due." E.g., see
http://en.wikipedia.org/wiki/Time_value_of_money,
http://www.had2know.com/finance/future-value-growing-annuity.html, etc.
From the wiki article: "For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due)." E.g. home mortgages are usually paid at the end of a period (and are thus "ordinary annuities") while investments are usually assumed to happen at the beginning (and are thus "annuities due").
Unless otherwise noted, an annuity formula is usually based on an ordinary annuity. You have to multiply by (1 + i), where i is the interest rate, to convert to an annuity due formula.
FV = Future Value
P = amount of the initial payment, $
N = number of payments (or number of years, etc.)
g = payment growth rate (decimal)
r = annual percentage rate (decimal)
FV = P[(1+r)^N - (1+g)^N]/(r-g) * (1+r) This is the "ordinary annuity" formula multiplied by (1+r) to get the "annuity due" formula.
For your example,
P = $10,000
N = 10
g = 0.25
r = 0.07
FV = $10,000 * (1.07^10 - 1.25^10)/(.07 - .25) * 1.07 = $436,683.
To check this, calculate the results of each year's investment. The first, $10,000, compounds at 7% for 10 years. The second, $12,500, compounds at 7% for 9 years, and so on.
| Investment | Years | 7% |
| $10,000 | 10 | $19,672 |
| $12,500 | 9 | $22,981 |
| $15,625 | 8 | $26,847 |
| $19,531 | 7 | $31,363 |
| $24,414 | 6 | $36,639 |
| $30,518 | 5 | $42,802 |
| $38,147 | 4 | $50,003 |
| $47,684 | 3 | $58,415 |
| $59,605 | 2 | $68,241 |
| $74,506 | 1 | $79,721 |
| | $436,683 |
Sorry - hope you weren't counting on the half billion dollars. ;)
