Increasing Annuity

An increasing annuity is an annuity whose payments rise over its term. Exam FM examines two patterns: payments that rise by a fixed amount each period (an arithmetic increasing annuity) and payments that rise by a fixed percentage (a geometric one). The standard symbols belong to the arithmetic family with payments $1, 2, 3, \ldots, n$ , one per period at the effective rate $i$ :

$(Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|} - nv^n}{i}$

$(Is)_{\overline{n}|} = \frac{\ddot{s}_{\overline{n}|} - n}{i}$

$(Ia)_{\overline{n}|}$ values payments $1, 2, \ldots, n$ at the ends of periods $1$ to $n$ , one period before the first payment; $(Is)_{\overline{n}|} = (1+i)^n\,(Ia)_{\overline{n}|}$ is their value at the last payment. The full treatment is Arithmetic Increasing Annuity.
Paid at the start of each period, the annuity-due versions divide by $d$ instead of $i$ : $(I\ddot{a})_{\overline{n}|} = (\ddot{a}_{\overline{n}|} - nv^n)/d$ and $(I\ddot{s})_{\overline{n}|} = (\ddot{s}_{\overline{n}|} - n)/d$ .
With no end date, the increasing perpetuity-immediate $1, 2, 3, \ldots$ is worth $(Ia)_{\overline{\infty}|} = 1/i + 1/i^2$ , and the perpetuity-due $(I\ddot{a})_{\overline{\infty}|} = 1/d^2$ .
A first payment $P$ rising by $Q$ a period is a level annuity of $P - Q$ plus $Q\,(Ia)_{\overline{n}|}$ — see Arithmetic Progression. The mirror image, payments $n, n-1, \ldots, 1$ , is the decreasing annuity $(Da)_{\overline{n}|}$ .
Payments growing by a constant percentage are the geometric increasing annuity and, with no end date, the geometric increasing perpetuity.

A Fund Built from Rising Deposits A fund receives deposits of $100 at the end of year 1, $200 at the end of year 2, and so on up to $1,000 at the end of year 10. It earns $5\%$ effective. Find its value just after the last deposit.
Answer The deposits are $100 \times (1, 2, \ldots, 10)$ , so the value is $100\,(Is)_{\overline{10}|}$ , with $s_{\overline{10}|} = 12.577893$ :
 $\begin{align*} \ddot{s}_{\overline{10}|} &= 1.05 \times 12.577893 \\ &= 13.206787 \\ (Is)_{\overline{10}|} &= \frac{13.206787 - 10}{0.05} \\ &= 64.13574 \\ AV &= 100 \times 64.13574 \\ &= 6{,}413.57 \end{align*}$ 
The deposits total $5,500; the other $913.57 is interest.

A Scholarship That Grows Forever A scholarship pays $1,000 at the end of the first year, $2,000 at the end of the second, $3,000 at the end of the third, and so on forever. At $i = 8\%$ , how much must be endowed today?
Answer
 $\begin{align*} PV &= 1{,}000\,(Ia)_{\overline{\infty}|} \\ &= 1{,}000\left(\frac{1}{0.08} + \frac{1}{0.08^2}\right) \\ &= 1{,}000\,(12.5 + 156.25) \\ &= 168{,}750 \end{align*}$ 
The level part alone, $1{,}000/0.08 = 12{,}500$ , is under a tenth of it: almost all the value is in the growth.