Need help with decreasing and increasing annuity in one question

[esmeralda1]

A guy has bought an annuity immediate starting with 1000$, decreasing every year by 50$ to finally reach 50$, then increasing every year by 50$ to reach back again to 1000$.
If the annual eff. rate of interest is 6%, what's the PV of this annuity?

The answer is 8724, but I find 8549.94

I divide it into first decreasing, then increasing annuity.
I use n, which is the 20 years to reach to 50$ in the first part (decreasing), then use n-1=19, which the time it takes to get back to 1000$ again by an increasing annuity, so the total time is 39 years.

I write the PV equation like this:
PV= Payment x ((Da)_n + v^n x (Ia)_n-1)

Payment= 50
(Da)_n= [20 -(a_20)]/0.06
v^n = 1.06^(-20)
(Ia)_n-1)= [1-(1.06^(-19)]/(0.06/1.06) - (19)[1.06(-19)]
---------------------------------------------
0.06

decreasing annuity part I calculate as 142.16798
increasing annuity part I calculate as 28.83079
I add the two parts=170.9988, then multiply it by the payment of 50.
The answer I find this way is 8549.94
Where do I make the mistake?

[Gandalf]

If you have access to Excel, often it is easier to figure out what has gone wrong by putting the cash flows into Excel, discounting each one, and adding them up. That would convince you that 8724 is the correct answer.

As to what went wrong: the increasing piece. You used a payment of 50 in year 21 (0 from the decreasing piece; 50 from the increasing piece). The payment in year 21 is 100. Similarly, all the remaining payments are 50 higher than you used.

[esmeralda1]

Thank you Gandalf.
I really do appreciate your help and explanations.

I reconsidered my timeline and payments match. You're right.
My increasing annuity starts at year 19 with 0 I assume, in year 20 the payment is 50, in year 21 payment is 100, then in year 39 it reaches 1000.
I'm rewriting my increasing annuity part again as follows:
discount factor = v^(-19) ...........since it's starting in year 19 with an imaginary 0.
(Ia)_n)= [1-(1.06^(-20)]/(0.06/1.06) - (20)[1.06(-20)]
The result comes out to be 32.62175507.
I add it up with the decreasing ann. factor 142.16798 to get a total of 174.7897351.
Then multiply it by 50, I get 8739.486754.
Can you confirm my solution please?

[Gandalf]

One way to do it (there are alternatives; I didn't really focus on what is going wrong in your second attempt).

A decreasing annuity immediate for 20 years is 142.16798 as you had.

An increasing annuity immediate for 19 years is 92.46427. A level immediate annuity for 19 years is 11.15812. v^20 = .311805

(142.16798+.311805*(92.46427+11.15812))*50 = 8723.90

Note that 11.15812*.311805*50 = 173.96. Add that to your first attempt and you're there.

[esmeralda1]

Thank you Gandalf.