Just write out the cash flow.
$$\begin{align*}
AV &= 100(1+i)^{19} + 100(1+i)^{18} + \cdots + 100(1+i)^{10} \\
&\hphantom{=} + 95(1+i)^9 + 90(1+i)^8 + \cdots + 55(1+i)^1 + 50.
\end{align*}$$
Note there are $19-10+1 = 10$ payments of $100$, and the remaining $9 - 0 + 1 = 10$ payments follow the arithmetic sequence $95, 90, \ldots, 50$.
Now you can see there are multiple ways of writing out this cash flow in actuarial notation. You could do it this way:
$$\begin{align*} AV
&= 100(1+i)^{10}\left((1+i)^9 + \cdots + 1 \right) \\
&\hphantom{=} + 45(1+i)^9 + 45(1+i)^8 + \cdots + 45 \\
&\hphantom{=} + 50(1+i)^9 + 45(1+i)^8 + \cdots + 5 \\
&= 100 (1+i)^{10} s_{\overline{10}\rceil i} + 45 s_{\overline{10}\rceil i} + 5(Ds)_{\overline{10}\rceil i}
\end{align*}$$
which is the textbook solution. What they did was take the first $10$ payments as an annuity-immediate of $10$ years on $100$, and accumulated it for an additional $10$ years; then split the decreasing annuity into a level annuity-immediate of $45$, and a decreasing annuity of $5$.
But you don't have to do it this way. You can do it a number of other ways, one of which is to write
$$\begin{align*}
AV &= 100\left((1+i)^{19} + \cdots + 1\right) \\
&\hphantom{=} - \left(5(1+i)^9 + 10(1+i)^8 + \cdots + 50\right) \\
&\hphantom{=} = 100s_{\overline{20}\rceil i} - 5(Is)_{\overline{10}\rceil i}.
\end{align*}$$
In both cases, the result is the same.
I have given this advice to numerous students of actuarial science: when in doubt, write out the cash flow. It takes practice and experience to be able to directly set up the equation of value in terms of actuarial notation, and when one is still learning, skipping the cash flow step is not only more error-prone, it may actually be more time-consuming than just writing it out because you are wasting time trying to come up with how to write it compactly.
There's no perpetuity here. Kimberley is offered a life annuity.
From the first option, we can compute the interest rate. In the second option, Kimberley is offered a life annuity with period certain, also worth $250,000$. A life annuity with a $10$-year period certain is the sum of a a ten-year annuity certain, and a 10-year deferred life annuity. Since ww know the interest rate, we can compute the value of the annuity certain, and then subtraction gives the value of the life annuity. We just have to divide by $13,000$ to get the answer to the question posed.
Best Answer
Your expression $$\int_{t=0}^{10} (1.06)^{-t} \, dt + \int_{t=10}^\infty (1.03)^{t-10} (1.06)^{-t} \, dt$$ is correct as written. If you did not get $27.028160912907478246\ldots$ then you have evaluated it incorrectly. Note that this can be written $$\int_{t=0}^{10} e^{-t \log 1.06} \, dt + (1.03)^{-10} \int_{t=10}^\infty e^{-t \log (1.06/1.03)} \, dt \\ = \frac{1}{\log 1.06} \int_{u=0}^k e^{-u} \, du + \frac{1}{(1.03)^{10} \log \frac{1.06}{1.03}} \int_{u=m}^\infty e^{-u} \, du,$$ where $k = 10 \log 1.06$ and $m = 10 \log \frac{1.06}{1.03}$.