Mathematics and Statistics · Ch 4 — Sequences and Series

Arithmetic Progression: nth Term

An Arithmetic Progression (AP) is a sequence in which each term after the first is obtained by adding a fixed number to the previous term. That fixed number is 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 whole AP is fixed: $a,\; a+d,\; a+2d,\; a+3d, \ldots$

Note

General (nth) Term of an AP

$t_n = a + (n-1)d$

To reach the $n$ -th term from the first term, add $d$ exactly $(n-1)$ times — not $n$ times, since the first term itself needs no addition.

Example: for $3, 7, 11, 15, \ldots$ the first term is $a = 3$ and $d = 7 - 3 = 4$ (checked: $11-7 = 4$ , $15-11 = 4$ — constant, so it genuinely is an AP). The $10$ th term is $t_{10} = 3 + (10-1)(4) = 3 + 36 = 39$ .

Note

$d$ May Be Negative or a Fraction …

Definition 3Common difference (d)

The constant amount added to each term of an AP to get the next: $d = t_{n+1} - t_n$ …

Definition 4nth term of an AP

$t_n = a + (n-1)d$ , where $a$ is the first term and $d$ the common …

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