In a general form, i know how to compute the PV of an annuity that follows a geometric progression.
But how do i compute it when it's the accumulated value?
let's say a payment of 100/ year with effective annual interest of 1% and increasing payments of 2% every year for 10 years.
The cash flow would be 100(1+i) + 100(1.02)(1+i)^2 + ... 100(1.02)^9 *(1+i)^10
we would have a common factor of (1.02)(1+i), how do i derive the formula from that?
For an annuity-immediate of $100$ per year increasing at $2\%$ per year with interest at $i = 0.01$, for a term of $n = 10$ years, the cash flow looks like this:
$$AV = 100(1.01)^9 + 100(1.02)(1.01)^8 + 100(1.02)^2(1.01)^7 + \cdots 100(1.02)^9.$$
Note I have written the terms in chronological order, so the first term $100(1.01)^9$ represents the accumulated value of the first payment, which because it is an annuity-immediate, occurs at the end of the first year of the $10$-year term. The second payment has increased to $100(1.02)$ but has had one less year to accumulate interest, so it has accumulated to $100(1.02)(1.01)^8$; and so on, until the last payment, which is $100(1.02)^9$ due to the geometric increase in payment, but as it is made at the end of the final year, has had no time to accrue interest.
Now note that since the general form of the $k^{\rm th}$ payment is $$100(1.02)^{k-1}(1.01)^{10-k},$$ we can write this as $$100(1.02)^9 (1.02)^{k-10}(1.01)^{10-k} = 100(1.02)^9 \left(\frac{1.01}{1.02}\right)^{10-k}.$$ So our accumulated value corresponds to a level annuity with modified interest rate; i.e., $$AV = 100(1.02)^9 s_{\overline{10}\rceil j},$$ where $j = \frac{1.01}{1.02} - 1 \approx -0.00980392$. You might wonder why the interest rate is negative, but it isn't; it's merely a convenient way to use the formula.
Alternatively, you could also write it as $$AV = 100(1.01)^9 s_{\overline{10}\rceil m},$$ where $m = \frac{1.02}{1.01} - 1 \approx 0.00990099$. It's the same thing. The point is that you can "absorb" the geometric progression in the payment into the interest rate because compound interest itself is effectively a geometric progression in the payment amount.
One final note. If you want to ask more questions on this site, I strongly advise you to learn how to use MathJax. Please see the tutorial here. In addition, you may learn how actuarial notation can be simulated by right-clicking (or control-clicking depending on your computer/browser) a mathematical expression, such as those in my answers. This opens up a small drop-down menu, the first option of which is "Show Math As." Select this, then select "TeX commands." This will open a pop-up window with the formula syntax that created the displayed expression.