Using $(I^{(m)}a)^{(m)}_{\bar{n}|}$ to solve for the present value of an annuity where payments increase monthly I've seen this answer and I understand the methodology, but I am wondering why my original solution using a different method did not work.
This is the sample problem in my study guide for Exam FM:

Olga buys a 5-year increasing annuity for X. Olga will receive 2 at the end of the first month, 4 at the end of the second month, and for each month thereafter the payment increases by 2. The nominal interest rate is 9% convertible quarterly. Calculate X.

$X$ is clearly the present value of this increasing annuity.
My study guide lists a formula for $(I^{(m)}a)^{(m)}_{\bar{n}|}$ as $$\frac{\ddot{a}^{(m)}_{\bar{n}|}-nv^n}{i^{(m)}}$$ This is supposed to be the present value of payments $\frac{1}{m^2}, \frac{2}{m^2}, \frac{3}{m^2},...,\frac{mn}{m^2}$ made at times $\frac{1}{m}, \frac{2}{m}, \frac{3}{m}, n$ respectively.
I initially thought the present value would be given by $288(I^{(12)}a)^{(12)}_{\bar{5}|}$ - this would give payments every month of $\frac{288}{144}=2, \frac{288*2}{144}=4, ...$. Applying the formula, however, gives me an answer that is far too large.
$$i \approx 9.3083\%$$
$$i^{(12)}\approx8.9333\%$$
$$\ddot{a}^{(12)}_{\bar{5}|} = \ddot{a}_{\bar{60}|0.7444\%} \approx 48.6083$$
$$288(I^{(12)}a)^{(12)}_{\bar{5}|} = 288\frac{48.6083-5(1.093083^{-5})}{0.08933} = 288(508.2575)=146378.16$$
The actual answer is $\approx 2729$.
Is this payment stream not accurately described using $(I^{(m)}a)^{(m)}_{\bar{n}|}$? If so why not? Admittedly, this is an overly-complex way to solve the problem compared to the linked answer but I'm hoping to better understand this type of increasing annuity.
 A: I find answering your question a bit hard since I don't know how you computed some terms. Anyway, here are some miskakes I see:

*

*Why do you multiply by $288$? You have an increase of $2$, so that is your multiplicative factor


*I don't get what $i$ and $i^{(12)}$ represent. Are they effective rates? Which ones? However in the formula for $\ddot{a}_{n|j}$ you used the correct rate, which is neither one of the two you wrote before.


*In the last formula, since we have monthly payments, you have $60$ periods and not $5$. Moreover in this formule you used the two rates I mentioned in point 2. but the rate you have to use is the same in all parts of the formula and doesn't change.
As a side note: your solution and the one linked are the same and use the same formula. The only difference, to me, is that you are a bit confused about which rate to employ.
To be explicit, I provide a possible solution with all the steps.
First, since we are given the nominal quarterly rate $i^{(4)}$, we need to convert it to the effective month rate $j$. The two are related from the formula
$$
(1+j)=\bigg(1+ \frac{i^{(4)}}{4}\bigg)^{4/12} \Rightarrow j = \bigg(1+ \frac{i^{(4)}}{4}\bigg)^{1/3} -1 = 0.007444
$$
Instead $v = (1+j)^{-1}$. Plug everything inside the formula and we end up with
$$
X = 2 \; \frac{\ddot{a}_{60|.007444} - 60 (1+.007444)^{-60}}{.007444} = 2729.21
$$
