POP

G4.7 The nth Term of a Linear Sequence

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01 — Explanation

What does the nth term formula actually represent?

For a linear (arithmetic) sequence — one where the same amount is added or subtracted every time — every term can be generated directly from its position $n$ using:

T_n = dn + b

where $d$ is the common difference (how much the sequence changes by each step) and $b$ is the zeroth term — the value the sequence would have at position $0$ , one step before the first term. This mirrors $y=mx+c$ : $d$ plays the role of gradient, $b$ the role of the $y$ -intercept.

02 — Explanation

How do I derive T_n from a list of numbers?

Find $d$ by subtracting consecutive terms: $d = T_2 - T_1$ .
Find $b$ by working backwards one step from the first term: $b = T_1 - d$ .
Write the final expression $dn+b$ .

Example (5, 7, 9, 11, …): $d=7-5=2$ . $b=5-2=3$ . $T_n=2n+3$ .

Example with negative terms (-7, -3, 1, 5, …): $d=-3-(-7)=4$ . $b=-7-4=-11$ . $T_n=4n-11$ .

Example with a decreasing sequence (15, 10, 5, 0, …): $d=10-15=-5$ . Since the operation was subtracting 5, finding $b$ means reversing that: $b=15-(-5)=20$ . $T_n=-5n+20$ .

Exam Tip

Sign errors when calculating $b$ are the single biggest source of lost marks here, especially when $T_1$ and $d$ have opposite signs. After finding your expression, always self-check by substituting $n=1$ and $n=2$ — if they regenerate the original first two terms, your formula is correct.

03 — Explanation

How do I check if a given number is a term in the sequence?

Set your $T_n$ formula equal to the target value $K$ and solve for $n$ :

dn+b=K

The result is only valid if $n$ is a positive whole number — a position in a sequence can't be the 287.5th term. If $n$ comes out as a decimal or negative, the number is not in the sequence, and you must state this explicitly as your conclusion, not just leave the decimal as the final answer.

Example: is 1143 a term of $T_n=4n-7$ ? $4n-7=1143 \Rightarrow 4n=1150 \Rightarrow n=287.5$ . Since this isn't a whole number, 1143 is not a term of the sequence.

04 — Explanation

What if the sequence is shown as a pattern of shapes, not numbers?

Count the relevant feature (dots, matchsticks, tiles) in each diagram first to convert it into a plain number sequence $T_1, T_2, T_3,\dots$ , then apply the usual $d$ -and- $b$ method exactly as above.

05 — Explanation

What's a common notation mistake to avoid?

If asked for "an expression, in terms of $n$ ," write only the expression itself (e.g. $4n-11$ ) — writing it as an equation like $n = 4n-11$ is algebraically wrong and can cost the final accuracy mark.

Exam Tip

Quadratic sequences (where the second difference is constant, rather than the first) are Higher Tier only — if the term-to-term differences aren't constant, don't try to force a linear $T_n$ onto it.

06 — Worked example

Worked example

The first four terms of a sequence are 91, 85, 79, 73. Find an expression for the nth term, and determine whether $-100$ is a term of the sequence.

$d = 85-91=-6$ .
$b = 91-(-6) = 97$ .
$T_n = -6n+97$ .
Set equal to $-100$ : $-6n+97=-100 \Rightarrow -6n=-197 \Rightarrow n=32.83\overline{3}$ .
Since $n$ is not a whole number, $-100$ is not a term of the sequence.