Arithmetic — a, d & the General Term

Section 1 of 3

Arithmetic Sequences

Add a fixed amount

An arithmetic (or linear) sequence goes up or down by the same amount each time. $a = T_1$ is the first term; $d = T_2 - T_1$ is the common difference.

Worked example — find a and d

For $3, 7, 11, 15, \ldots$ find $a$ and $d$.

$a = T_1 = 3$

$d = T_2 - T_1 = 7 - 3 = 4$

$a = 3,\ d = 4$

You try

Find $a$ and $d$ for (i) $5, 11, 17, 23, \ldots$; (ii) $7, 3, -1, -5, \ldots$; (iii) $-9, -6, -3, \ldots$

$a = T_1$, $d = T_2 - T_1$.

(i) $a = 5,\ d = 6$

(ii) $a = 7,\ d = 3 - 7 = -4$

(iii) $a = -9,\ d = -6 - (-9) = 3$

see steps

Section 2 of 3

The General Term

The formula

$T_n = a + (n-1)d$ (from the Tables). Multiply out and tidy up to get a rule for any term.

Worked example — build the rule

Find $T_n$ for $2, 5, 8, 11, \ldots$

$a = 2$, $d = 3$

$T_n = 2 + 3(n-1) = 2 + 3n - 3$

$T_n = 3n - 1$

Worked example — a decreasing sequence

Find $T_n$ for $18, 15, 12, 9, \ldots$

$a = 18$, $d = -3$

$T_n = 18 - 3(n-1) = 18 - 3n + 3$

$T_n = 21 - 3n$

Section 3 of 3

Practice

You try

Find $T_n$ for $7, 11, 15, 19, \ldots$

$a = 7$, $d = 4$.

$T_n = 7 + 4(n-1) = 4n + 3$

$T_n = 4n + 3$

You try

Find $T_n$ for $25, 31, 37, 43, \ldots$

$a = 25$, $d = 6$.

$T_n = 25 + 6(n-1) = 6n + 19$

$T_n = 6n + 19$

You try

Find $T_n$ for $7, 2, -3, -8, \ldots$

$a = 7$, $d = -5$.

$T_n = 7 - 5(n-1) = 12 - 5n$

$T_n = 12 - 5n$

You try

Find $T_n$ for $-12, -16, -20, \ldots$

$a = -12$, $d = -4$.

$T_n = -12 - 4(n-1) = -4n - 8$

$T_n = -4n - 8$

That’s Class 2.

$a$, $d$ and the general term $T_n = a + (n-1)d$. Class 3: using the rule to find a term or which term.