Business Mathematics and Basic Statistics · Ch 7 — Arithmetic and Geometric Progressions

Arithmetic Progression: First Term and Common Difference

An Arithmetic Progression (AP) is a sequence $T_1, T_2, T_3, \ldots$ in which the difference between any term and the one immediately before it is always the same constant. This constant is called the common difference, denoted $d$ :

d = T_2 - T_1 = T_3 - T_2 = T_4 - T_3 = \cdots

The first term is denoted $a$ (so $a = T_1$ ). Once $a$ and $d$ are known, the entire AP is determined — every subsequent term is obtained by repeatedly adding $d$ : $a,\; a+d,\; a+2d,\; a+3d,\ldots$

Example: in the AP $5, 11, 17, 23, \ldots$ , the first term is $a=5$ and the common difference is $d = 11-5 = 6$ (checked again: $17-11=6$ , $23-17=6$ — constant, confirming this genuinely is an AP).

Note

$d$ Can Be Negative or a Fraction

A common difference is not required to be a positive whole number. The sequence $20, 17, 14, 11, \ldots$ is an AP with $a=20$ and $d=-3$ (a decreasing AP — this is exactly the pattern behind straight-line depreciation, where a fixed amount is subtracted from the book value every year). Likewise $d$ may be a fraction, e.g. $2, 2.5, 3, 3.5,\ldots$ has $d=0.5$ . …

Definition 2Common difference (d)

The constant amount added to each term of an Arithmetic Progression to obtain the next term: $d = T_{n+1}- …

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