The question asks: 'A 35-year annuity immediate pays $1.05^{35}$ in the first year, $1.05^{34}$ in the second year, etc., until 1.05 is paid in the 35th year. The PV of this annuity at 5% effective is X. Determine X.'
I understand the answer = $1.05^{34} + 1.05^{32} + ... + 1.05^{-32} + 1.05^{-34}$. However I don't get why it can be simplified to [(s-angle-34)+(a-angle-36)]/(a-angle-2)?
Please help! Thanks in advance!
To be honest, don't get too hung up on trying to find equivalent ways to write the cash flow in terms of actuarial notation. I would go about it this way: We know that the series is geometric with common ratio $1/(1.05)^2 = v^2$, where $v$ is the annual present value discount factor. We also know that this series begins at $(1.05)^{34} = v^{-34}$ and has a term of $35$ years. So $$PV = \frac{v^{-34}(1-(v^2)^{35})}{1-v^2}.$$ And at this point, you can already do the calculation. Now if you really had your heart set on writing it in actuarial notation, we would do some algebra: $$\begin{align*} PV &= \frac{v^{-34} - v^{36}}{1-v^2} \\ &= \frac{(1+i)^{34} - 1 + 1 - v^{36}}{i} \cdot \frac{i}{1-v^2} \\ &= \frac{s_{\overline{34}\rceil} + a_{\overline{36}\rceil} }{a_{\overline{2}\rceil}}, \end{align*}$$ as claimed. Note that we used a little trickery, subtracting and adding 1 in the numerator, and then multiplying/dividing by $i$ in the denominator, to get things to match up with the formulas we know for $a$ and $s$. The reason why I don't recommend this approach is for two reasons:
Now, doing this extra work could be a useful tool on the exam if you had extra time left over to do it. You could use it as a way to check the validity of your calculations. But the point here is to never make more effort to solve a question than is absolutely required. Remember the basic formulas, but also don't rely on those formulas too much: use your common sense.