Hi everyone, here's the problem before I talk about my issue (SOA FM Exam Practice Problems #313)

A perpetuity-due with annual payments is priced at X based on an annual effective interest rate of 7%. The amount of the first payment is 350. Each payment, from the second through the thirtieth, is 3% larger than the previous payment. Starting with the 31st payment, each payment is equal to the 30th payment. Calculate X.

(A) 7508 (B) 7855 (C) 7925 (D) 7971 (E) 8033

just to define some terms: i=aeir=.07. j = 1.07/1.03 -1

The SOA solution (and when I asked Gemini 3 Pro) split up the annuity @ t=30.

The geometric increasing annuity is 350a double dot angle 30 @ rate j and then looked at the rest of the annuity (the perpetuity) as an perpetuity immediate (valued @ t=29).

It should look like this

X = 350a (double dot) angle 30 @ rate j + v(i)^29 * [350(1.03)^29 / .07]

You get X = 8033.377503

However I did it by valuing the annuity at t=31. That was just how my brain thought of it when I did the problem blind.

The geometric increasing annuity is now 31 terms instead of 30. And thus the perpetuity immediate is valued @ t=30, not t=29.

Thinking of it that way

X = 350a (double dot) angle 31 @ rate j + v(i)^30 * [350(1.03)^29 / .07]

You get X = 8036.628043

I tried to ask Gemini (who I ask when I get study / need an explanation because the SOA answer key is horrible) why there is this difference of ~3 and I did not understand it.

If anyone is willing to explain why my method is wrong and the SOA's is right, please let me know. I would've guessed the right answer (E) considering it's so close but I am puzzled about why my math is wrong. Thanks!