I need help figuring out recurrence relations for various annuities. I've attacked the questions below and my responses. I'm not too sure what the recurrence relation is for d) however. I have attached what I have done so far for all the recurrence relation questions. Any help is much appreciated
Questions:
A recurrence relation is an equation that recursively defines a sequence of values, whereby
each element of a sequence can be written as a function of preceding element(s);
the first element of the sequence will be uniquely defined by an initial value of the recurrence relation.
Specifically, if a sequence $u_n$ can be expressed as a function of only n and the immediate preceding element $u_n−1$, i.e., $u_n = g(n, u_n−1)$, then we say that $u_n$ is a recurrence relation of order 1. The values of the entire sequence can be calculated recursively starting from the initial value say $u_1$ and then by $u_2 = g(2, u_1)$ and more generally $u_n = g(n, u_n−1)$ for $n = 3, 4, 5, · · · $
(a) Write down a recurrence relation for $\ddot a_n$ . Explain your thought process in words (e.g.,using the timeline approach) OR prove the result mathematically by first principles. Also write down the initial value for the sequence.
My answer-
Recurrence relation: $\ddot a_ {\bar n|} = \ddot a_{\bar {n-1}|} + v^{n-1}$
$\ddot a_ {\bar1|}=1$
$\ddot a_ {\bar n|}=1+v+v^2+...+v^{n-1}$
$\ddot a_ {\bar 2|} =1+v= \ddot a_1+v$
Therefore the regression is proven
(b) Given an effective discrete periodic rate of 3% per period, tabulate the values of $\ddot a_n$ for $n = 1, 2, · · · , 30.$
(c) Repeat parts (a) and (b) for an instead of $\bar a_n $ .
My recurrence relation: $ \bar a_n = \bar a_{n-1} + \frac {d} {\delta} v^{n-1}$
Initial value: $ \bar a_1 = \frac {d} {\delta}$
$ \bar a_2 = \frac {d} {\delta} + \frac {d} {\delta} v = \bar a_1 + \frac {d} {\delta} v$
And so on therefore the regression is proven
(d) Repeat parts (a) and (b) for $(I \ddot a)_n$ instead of $\ddot a_n$ .
$\ddot a_{\overline{n}\rceil i}$ represents the present value of an annuity-due of $1$ paid at the beginning of each period with periodic effective interest rate $i$, for $n$ periods (payments). As such, it is easy to see that the present value of the $k^{\rm th}$ payment is $v^{k-1}$, where $v = 1/(1+i)$ is the periodic present value discount factor. The sum of the present values of each payment is therefore $$\ddot a_{\overline{n}\rceil i} = 1 + v + v^2 + \cdots + v^{n-1}.$$
We may also observe that for such an annuity-due, if there is an additional payment at the beginning of period $n+1$, the present value of this payment is simply $v^n$; therefore, the total present value is the sum of the present values of the annuity-due for the first $n$ periods, plus the present value of the $(n+1)^{\rm th}$ payment; i.e. $$\ddot a_{\overline{n+1}\rceil i} = (1 + v + v^2 + \cdots + v^{n-1}) + v^n = \ddot a_{\overline{n}\rceil i} + v^n.$$ This is one such recursion relation.
Another possible recursion relation that we may write is found by considering that by deferring an annuity-due of $n$ payments by one period, we get the equivalent of the cash flow for payments $2, 3, \ldots, n+1$ on an annuity-due with $n+1$ payments, hence $$\ddot a_{\overline{n+1}\rceil i} = 1 + v(1 + v + v^2 + \cdots + v^{n-1}) = 1 + v \ddot a_{\overline{n}\rceil i}.$$
Part (b) is trivial and is left as an exercise.
Part (c) is unclear. Is the question asking for the recurrence for $a_{\overline{n}\rceil i}$, an annuity-immediate for $n$ periods, or for $\bar{a}_{\overline{n}\rceil i}$, a continuous annuity over $n$ periods with effective periodic interest rate $i$?
Part (d) requires more consideration than you have provided in your work. We have $$(I\ddot a)_{\overline{n}\rceil i} = 1 + 2v + 3v^2 + \cdots + nv^{n-1}.$$ This is the definition along the lines of what we have reasoned for the level annuity-due above. We wish to establish a recursion relation for this. Of course, we can simply write $$(I\ddot a)_{\overline{n+1}\rceil i} = (1 + 2v + 3v^2 + \cdots + nv^{n-1}) + (n+1)v^n = (I \ddot a)_{\overline{n}\rceil i} + (n+1)v^n,$$ or we can write $$\begin{align*} (I \ddot a)_{\overline{n+1}\rceil i} &= 1 + 2v + 3v^2 + \cdots + (n+1)v^n \\ &= (1 + v + v^2 + \cdots + v^n) + (v + 2v^2 + 3v^3 + \cdots + nv^n) \\ &= \ddot a_{\overline{n}\rceil i} + v(1 + 2v + 3v^2 + \cdots + nv^{n-1}) \\ &= \ddot a_{\overline{n}\rceil i} + v (I\ddot a)_{\overline{n}\rceil i} \end{align*}$$ The problem you are having is that you are not reasoning from first principles, instead electing to use the derived formulas for these annuities, when you have not in fact demonstrated them to be true. Moreover, these formulas are not particularly useful for writing out recurrence relations, since the original summation much more readily yields a number of possible recurrences.