Common Patterns

Why the Common Patterns are Important

By recognizing the common patterns, complex problems can be analyzed by breaking them up into a series of smaller problems, and the end solution can be reassembled from the individual solutions.


  • A single present or future cash flow.
  • Compound Amount: (F|P,i,N), $F = P(1+i)^N$
  • Present Worth: (P|F,i,N), $P = F(1+i)^{-N}$

Constant Series2

A series of flows of equal amounts at regular intervals for periods 1 through N.

Here are some examples of how to calculate present worth and future worth of a constant series by brute force — the fall back method. Please keep in mind that brute force will get you an answer, but it is hard to spot when you have made a mistake and it can be very time consuming for long series.

The following equations are developed in this video. Please note that the technique can be used to develop the other present worth and future worth equations.

  • Compound Amount: (F|A,i,N), $F = A [((1+i)^N-1)/i]$
  • Sinking Fund: (A|F,i,N), $A = F [i/((1+i)^N-1)]$
  • Present Worth: (P|A,i,N), $P = A [((1+i)^N-1)/(i(1+i)^N)]$
  • Capital Recovery: (A|P, i, N), $A = P[i(1+i)^N/((1+i)^N-1)]$

Midterm 1 Constant Series Questions

Linear Gradient Series3

A series of flows the increases or decreases by a set amount, G, starting at zero in time period one.

  • Present Worth: (P|G,i,N), $P = G [((1+i)^N-iN-1)/(i^2(1+i)^N)]$
  • Conversion Factor: (A|G,i,N), $A = G[((1+i)^N-iN-1)/(i[(1+i)^N-1])]$

Geometric Gradient Series4

A series of flows that increases or decreases by a fixed percentage at regular intervals starting with a non-zero value in period one.

  • Present Worth: (P|A1, g,i,N), $P = A1[(1-(1+g)^N(1+i)^{-N})/(i-g)]$ or $P = A1(N/(1+i))$ if i=g
  • NOTE: The tables can not be used for geometric gradient!


A series of flows that does not exhibit an overall pattern. Patterns can sometimes be detected within the irregular pattern allowing it to be subdivided into one of the more familiar patterns.


The following represent good sample questions that would help you prepare for an exam:

Common patterns Question 1
Common patterns Question 2
Common patterns Question 3

NEXT: Decomposition

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