Arithmetic gradient series

An arithmetic gradient series is a cash-flow pattern that starts at zero in period 1, grows by a constant amount $G$ each period, and reaches $(N-1)G$ in period $N$. The cash flow in period $t$ is $(t-1)G$:

Period	1	2	3	…	$N$
Cash flow	0	$G$	$2G$	…	$(N-1)G$

Two factors handle the standard equivalences:

Arithmetic gradient to present worth.

$P=G\cdot (P/G,i,N),(P/G,i,N)=\frac{(1+i{)}^N-iN-1}{i^2(1+i{)}^N}.$

Arithmetic gradient to annuity.

$A=G\cdot (A/G,i,N),(A/G,i,N)=\frac{1}{i}-\frac{N}{(1+i{)}^N-1}.$

The first gives the present worth of the whole gradient stream; the second converts the gradient into the equivalent annuity of equal payments (different magnitude than the gradient’s terms, but same total present worth).

When cash flows look like $A_1,A_1+G,A_1+2G,\dots$ — a constant base $A_1$ plus an arithmetic gradient — decompose into two pieces: a uniform annuity of $A_1$ and a pure arithmetic gradient with first term $0$ and step $G$. Then combine:

$P=A_1\cdot (P/A,i,N)+G\cdot (P/G,i,N).$

Engineering applications: an asset’s maintenance cost rises by a roughly constant amount each year as it ages (\0$year1,$$200$year2,$$400$ year 3, …); a contract whose payments step up by a fixed annual amount; a depreciation pattern in some accounting conventions. Whenever the change per period is constant, an arithmetic gradient fits.

For a multiplicative version (cost grows by a constant percentage each period) see Geometric gradient series. For the underlying factor family see Compound interest factor.