Decreasing perpetuity problem A perpetuity pays 1000 immediately. The second payment is 97% of the first payment and is made at the end of the fourth year. Each subsequent payment is 97% of the previous payment and is paid four years after the previous payment. Calculate the present value of this annuity at an annual effective rate of 8%.
My attempt:
Let X denote the present value.
$x/(1+i) + 0.97x/(1+i)^5 + 0.97^2x/(1+i)^9 + ...$ = 1000
where $i = 0.08$, however I am not sure how to solve this or if this is correct

Since 1000 is paid immediately the present value will be equal to 1000 plus the NPV of the stream of future payments.
The dollar value of the nth future payment is $1000\cdot 0.97^n$
Each future payment is discounted by a factor of $\frac1{1.08^{4n}}$ since payments are every 4 years
So $$NPV = 1000 + \sum_{n=1}^\infty\frac{1000\cdot 0.97^n}{1.08^{4n}}=
1000 + 1000\cdot \sum_{n=1}^\infty\biggl(\frac{0.97}{1.08^4}\biggr)^n$$
$$=1000 + 1000\cdot \sum_{n=1}^\infty0.71298^n=1000\cdot \sum_{n=0}^\infty0.71298^n$$
$$=1000\cdot \frac1{1-0.71298}=3484.07$$ using $$\sum_{n=0}^\infty\alpha^n=\frac1{1-\alpha}$$
