In the ASM manual for exam FM we are given the following identity
$$(Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)a_{\overline{n}|}$$
however I am unable to prove it.
It may seem like a dumb question to ask but I am just not having any luck as to finding the trick to make them equivalent. I am writing Exam FM in June and I want to work on my understanding of the formulas so that if I get stuck on the exam, I know multiple ways to convert formulas and to use the formulas I remember to answer the questions.
Assume an annual interest rate of $i$, with payments occurring at years/times $1$, $2$, $\dots$, $n$.
$(Ia)_{\overline{n}|}$ is defined as the present value of payments $1, 2, \dots, n$ at these corresponding times.
$(Da)_{\overline{n}|}$ is defined as the present value of payments $n$, $n-1$, $\dots$, $1$ at these corresponding times.
Thus, $(Ia)_{\overline{n}|} + (Da)_{\overline{n}|}$ is the present value of $1+n$, $2+(n-1)$, $\dots, $ $n+1$ or payments of $n+1$ over $n$ years, hence $(Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)a_{\overline{n}|}$.
If this isn't satisfactory enough for a proof for you, the math isn't much more insightful: $$ \begin{align} (Ia)_{\overline{n}|} &= 1v+2v^2+\cdots+nv^n \\ (Da)_{\overline{n}|} &= nv+(n-1)v^2+\cdots+1v^{n} \\ (Ia)_{\overline{n}|}+(Da)_{\overline{n}|} &= (n+1)v+(n+1)v^2+\cdots+(n+1)v^n \\ &= (n+1)[v+v^2+\cdots+v^n] \\ &= (n+1)a_{\overline{n}|}\text{.} \end{align}$$