Arithmetic Gradient Series Calculator (P/G and A/G Factors)
Find the present value, future value and equivalent uniform series of cash flows that grow (or shrink) by a fixed amount each period, using the engineering-economics P/A, P/G and A/G factors.
How it works
Cash flow in period t = A₁ + (t − 1) × G · P = A₁ × (P/A, i, n) + G × (P/G, i, n) · (P/A) = [(1 + i)ⁿ − 1] ÷ [i (1 + i)ⁿ] · (P/G) = [(1 + i)ⁿ − i n − 1] ÷ [i² (1 + i)ⁿ] · (A/G) = 1 ÷ i − n ÷ [(1 + i)ⁿ − 1] · A = A₁ + G × (A/G) · F = P × (1 + i)ⁿ
An arithmetic (linear) gradient is a cash-flow series that changes by the same amount every period — maintenance costs that rise by 100 a year, or rent income that falls by a fixed amount as a building ages. Engineering-economics texts split it into a level series equal to the first cash flow plus a pure gradient that starts at zero in period 1 and grows by G per period; the P/G and A/G factors convert that gradient to a present value or an equivalent uniform series. A series that grows by a fixed percentage instead is a geometric gradient (growing annuity).
Worked example
Cash flows of 1,000 in period 1 that rise by 100 each period for 10 periods (up to 1,900) at 8%: (P/A, 8%, 10) = 6.7101 and (P/G, 8%, 10) = 25.9768, so P = 1,000 × 6.7101 + 100 × 25.9768 = 9,307.76. The equivalent uniform series is 1,000 + 100 × 3.8713 = 1,387.13 per period, and the future value after 10 periods is 9,307.76 × 1.08¹⁰ = 20,094.77.
Sources: Wikipedia — Engineering economics · Wikipedia — Time value of money
Frequently asked questions
What is an arithmetic gradient?
A series of cash flows that increases or decreases by a constant amount G each period. The first period’s flow is the base amount A₁ and the gradient starts adding in period 2.
What is the difference between an arithmetic and a geometric gradient?
An arithmetic gradient changes by a fixed amount (for example +100 per year); a geometric gradient changes by a fixed percentage (for example +5% per year), which is valued with the growing annuity formula.
How do I handle a decreasing series?
Enter a negative gradient. The same formulas apply: P = A₁ × (P/A) − |G| × (P/G).
What does the equivalent uniform series mean?
It is the constant amount per period that has the same present value as the changing series — useful for comparing a rising cost stream with a fixed payment.
Related calculators
Educational estimate, not financial advice. Real contracts, fees, taxes, day-count and rounding rules differ — check the official cost disclosure and your agreement before deciding.