7.3Finding the General Term of a Linear Sequence

A linear sequence goes up by the same amount each time. That amount is the common difference $d$ , and the term you would have had at position $0$ is $T_0$ .

General term of a linear sequence

\displaystyle T_n = dn + T_0 \quad \text{where } d = T_2 - T_1

Reading off

d

and

T_0

from a linear sequence

\displaystyle \begin{array}{ccccccc} \text{Position } n: & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Term } T_n: & 7 & 11 & 15 & 19 & 23 \\ & & +4 & +4 & +4 & +4 \end{array}

Common difference $d = 4$ and first term $7$ , so $T_0 = 7 - 4 = 3$ , the $0^{\text{th}}$ term, and $T_n = 4n + 3$ .

Find $d = T_2 - T_1$ , and check it against $T_3 - T_2$ .
Write $T_n = dn + T_0$ .
Find $T_0 = T_1 - d$ , the $0^{\text{th}}$ term: one step back from the first.
Verify with $n = 1$ and $n = 2$ .

The terms of

7, 11, 15, 19, 23

plotted against

n

Finding the formula: positive common difference

Find a formula for $\displaystyle T_n$ and hence find the 40th term: $\displaystyle 9, 16, 23, 30, 37, \ldots$

Show solution▾

Common difference

d = 16 - 9 = 7

\displaystyle T_n

\displaystyle = 7n + c

\displaystyle T_0

\displaystyle = T_1 - d = 9 - 7 = 2

\displaystyle T_n

\displaystyle = 7n + 2

Check:

T_1 = 7(1) + 2 = 9

✓,

T_2 = 7(2) + 2 = 16

✓

\displaystyle T_{40}

\displaystyle = 7(40) + 2

\displaystyle = 280 + 2

\displaystyle = 282

Finding the formula: sequence with negative terms

Find a formula for $\displaystyle T_n$ : $\displaystyle -11, -5, 1, 7, 13, \ldots$

Show solution▾

Common difference

d = -5 - (-11) = 6

\displaystyle T_n

\displaystyle = 6n + c

\displaystyle T_0

\displaystyle = T_1 - d = -11 - 6 = -17

\displaystyle T_n

\displaystyle = 6n - 17

Check:

T_1 = 6(1) - 17 = -11

✓,

T_3 = 6(3) - 17 = 1

✓

Finding the formula: negative common difference

Find a formula for $\displaystyle T_n$ : $\displaystyle 85, 72, 59, 46, 33, \ldots$ Then determine whether 5 appears in this sequence.

Show solution▾

Common difference

d = 72 - 85 = -13

\displaystyle T_n

\displaystyle = -13n + c

\displaystyle T_0

\displaystyle = T_1 - d = 85 - (-13) = 98

\displaystyle T_n

\displaystyle = -13n + 98

Check:

T_1 = -13(1) + 98 = 85

✓,

T_2 = -13(2) + 98 = 72

✓

To check if 5 appears: set

-13n + 98 = 5

\displaystyle -13n

\displaystyle = -93

\displaystyle n

\displaystyle = \frac{93}{13}

n \approx 7.15

which is not a whole number.

Therefore 5 does not appear in this sequence.

Now you try

Find $T_n$ for the sequence $5, 8, 11, 14, 17, \ldots$

Show solution▾

5, 8, 11, 14, 17, \ldots

\displaystyle \text{Common difference } d

\displaystyle = 8 - 5 = 3

\displaystyle T_n

\displaystyle = 3n + c

\displaystyle T_0

\displaystyle = T_1 - d = 5 - 3 = 2

\displaystyle T_n

\displaystyle = 3n + 2

Want it marked, with hints, and a new question each time? Practise this →