Present value of varying annuities I need help with the following questions. In order to answer these questions, I am only allowed to use two of the tables of values for $\ddot a_n , \bar a_n$ (continuous annuity) and   $(I \ddot a)_n$ from n=1,...,30 and i=3%.
(h) Given an effective annual rate of 3%, calculate the present value at time 0 of a 15-year arithmetically increasing annuity immediate whereby the first annual payment is 2,452
and subsequent annual increment is 450. 
$PV_0 = 2002 \ddot a_{15} + 450 (I \ddot a)_{15}$
$ = 2002*12.29607 + 450*91.60627$
=$65,839.55
(i) Consider a stream of 30-year continuous cash flows with a payment rate of 404t during year t. For instance, the payment rate during year 5 is $2,020. Calculate its present value
at time 0 given an effective annual rate of 3%.
$PV_0 = 404(I \bar a)_{30} + 404 \bar a_{30}$
$= 404(I \ddot a)_{30}* \frac{d}{\delta} + 404 \bar a_{30}$
$=404*268.7906* \frac {1-(1.03)^{-1}}{ln(1.03)} +404*19.893$
=$115,038.96
Any help is appreciated, especially for i) as I'm not too sure about that question.
Thanks
 A: You need to write out the cash flows and set up the equation of value.
For (h), an increasing annuity-immediate has the first payment of $2452$ at time $t = 1$, so the equation of value is $$PV = 2452 v + 2902 v^2 + \cdots + 8752 v^{15}$$ where $v = 1/(1+i)$ and $i = 0.03$.  Using your line of reasoning, this would be $$PV = 2002(v + v^2 + \cdots + v^{15}) + 450(v + 2v^2 + 3v^3 + \cdots 15v^{15}).$$  Therefore, the correct equation in actuarial notation would need to be $$PV = v\left(2002 \ddot a_{\overline{15}\rceil i} + 450 (I \ddot a)_{\overline{15}\rceil i}\right).$$
For (i), the equation of value is more difficult to express.  Consider the cash flow in year $t$ alone.  This corresponds to 
$$404t {\bar a}_{\overline{1} \rceil i}$$
before discounting for the year in which the payment is made.  In other words, we are only taking the present value of the payments in year $t$ to time $t-1$.  Now if we discount for the year, this is an extra factor of $v^{t-1}$.  So the cash flow is 
$$PV = 404 {\bar a}_{\overline{1}\rceil i} + 808 v {\bar a}_{\overline{1}\rceil i} + \cdots + 12120 v^{29} {\bar a}_{\overline{1}\rceil i}.$$  Factoring, we get $$PV = 404 {\bar a}_{\overline{1}\rceil i} \left( 1 + 2v + 3v^2 + \cdots + 30v^{29} \right) = 404 {\bar a}_{\overline{1}\rceil i} (I\ddot a)_{\overline{30}\rceil i}.$$
