What I think you are looking for is the present value of a growing annuity.
- Your problem says that your nominal interest rate is 6%, you will have to find the effective interest rate:
$$r=(1+\frac{r_n}{12})^{12}-1$$
$$0.061678=(1+\frac{0.06}{12})^{12}-1$$
- Then you can find the present value of the annuity:
$$PV=C\left(\frac{1}{r-g}\right)\left(1-\left(\frac{1+g}{1+r}\right)^N\right)$$
$$429607.68=2000\left(\frac{1}{(\frac{0.061678}{12})-(\frac{0.05}{12})}\right)\left(1-\left(\frac{1+\frac{0.05}{12}}{1+\frac{0.061678}{12}}\right)^{240}\right)$$
I still don't get the same answer than your book. However, I didn't try converting everything to a yearly payment. Calculate it rather than on 240 \$2000 payments, on 20 \$x payments.
The way you describe the cash flow does not match the equation of value you wrote.
You say that the payments are increased by $50$ for each year for the first ten years, and then by $100$ for each year for the last ten years. But the way you wrote it, you're increasing the payments by $150$ in the last ten years. Which is it?
My interpretation of the question would have the payments be:
$$1000, 1050, 1100, 1150, 1200, 1250, 1300, 1350, 1400, 1450, \\
1550, 1650, 1750, 1850, 1950, 2050, 2150, 2250, 2350, 2450.$$ Note that the total increase from the first payment to the last is $50(9) + 100(10) = 1450 = 2450 - 1000$.
In such a case, there are multiple ways to write the equation of value. One way is to take the level payments of $1000$ out, and then what is left are two increasing annuities, one deferred by one year, and the other deferred by $10$ years. Refer to the following table:
$$\begin{array}{c|c|c|c|c}
\text{Year} & \text{Level Payment} & \text{Increasing 1} & \text{Increasing 2} & \text{Total} \\
\hline
1 & 1000 & 0 & 0 & 1000 \\
2 & 1000 & 50 & 0 & 1050 \\
3 & 1000 & 100 & 0 & 1100 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
10 & 1000 & 450 & 0 & 1450 \\
11 & 1000 & 500 & 50 & 1550 \\
12 & 1000 & 550 & 100 & 1650 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
20 & 1000 & 950 & 500 & 2450 \\
\end{array}$$
This gives us the equation of value
$$\require{enclose}
\begin{align}
PV
&= 1000(v + v^2 + \cdots + v^{20}) + (50v^2 + 100v^3 + \cdots + 950v^{20}) \\ &\qquad + (50v^{11} + 100v^{12} + \cdots + 500v^{20}) \\
&= 1000 a_{\enclose{actuarial}{20} i} + 50 v (Ia)_{\enclose{actuarial}{19}i} + 50 v^{10}(Ia)_{\enclose{actuarial}{10}i}.
\end{align}$$
Alternatively, we can structure the payments as follows:
$$\begin{array}{c|c|c|c|c}
\text{Year} & \text{Level Payment} & \text{Increasing 1} & \text{Increasing 2} & \text{Total} \\
\hline
1 & 950 & 50 & 0 & 1000 \\
2 & 950 & 100 & 0 & 1050 \\
3 & 950 & 150 & 0 & 1100 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
10 & 950 & 500 & 0 & 1450 \\
11 & 1450 & 0 & 100 & 1550 \\
12 & 1450 & 0 & 200 & 1650 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
20 & 1450 & 0 & 1000 & 2450 \\
\end{array}$$
This gives the equation of value
$$PV = 950 a_{\enclose{actuarial}{10}i} + 1450 v^{10} a_{\enclose{actuarial}{10}i} + 50 (Ia)_{\enclose{actuarial}{10}i} + 100v^{10} (Ia)_{\enclose{actuarial}{10}i}.$$
You might ask, why would we write it this way instead of the previous? Well, notice that even though there is one more annuity symbol, there's actually only two different ones, rather than three in the first equation of value. In other words, things factor more nicely in the second equation:
$$PV = 50\left( (19 + 29v^{10})a_{\enclose{actuarial}{10}i} + (1 + 2v^{10})(Ia)_{\enclose{actuarial}{10}i}\right).$$
In both cases, the answer is the same, approximately $11843.639297$. But no matter how you write the equation of value, you must use an increasing annuity symbol $(Ia)_{\enclose{actuarial}{n}i}$.
Best Answer
The cash flow of payments into the fund is finite. The cash flow of income paid by the fund is not; it is a perpetuity because it must be sustaining for as long as he survives.
In actuarial notation, we have $$X \ddot s_{\overline{300}\rceil j} = 3000 \ddot a_{\overline{\infty}\rceil k},$$ where $j = i^{(12)}/12 = 0.08/12$, and $k = 9.65/1000 = 0.00965$. This equation of value is written with respect to the time of the first receipt of income from the fund. Since $$\ddot s_{\overline{n}\rceil j} = (1+j) \frac{(1+j)^n - 1}{j}, \\ \ddot a_{\overline{\infty}\rceil k} = 1 + \frac{1}{k},$$ it follows that $$X = 3000 \cdot \frac{104.627}{957.367} = 327.858\ldots.$$