Think First
warm-up

Before diving into the practice questions, take a moment to think about what you already know about this topic.

Record in workbook.
1
What You'll Master
objectives

Know

  • Arithmetic sequences have a fixed common difference $d$ between consecutive terms
  • $d$ can be found by calculating $T_2 - T_1$
  • The first term is labelled $a$

Understand

  • Each term can be expressed using the nth-term formula $T_n = a + (n-1)d$
  • This formula always simplifies to a linear expression of the form $T_n = dn + c$
  • If substituting a value gives a non-integer $n$, that value is not in the sequence

Can Do

  • Find the common difference of any arithmetic sequence
  • Write the nth-term rule for a given sequence
  • Find any specific term or test whether a value belongs to a sequence
2
Words You Need
vocabulary
SequenceAn ordered list of numbers that follow a rule. Each number in the list is called a term.
TermA single number in a sequence. The first term is $T_1$, the second is $T_2$, and the $n$th term is $T_n$.
Common Difference ($d$)The fixed amount added (or subtracted) between consecutive terms in an arithmetic sequence. $d = T_2 - T_1$.
First Term ($a$)The starting value of a sequence, labelled $a$ or $T_1$. It is the value when $n = 1$.
nth-Term FormulaA rule that gives the value of any term directly: $T_n = a + (n-1)d$. Substitute any $n$ to find that term.
Arithmetic SequenceA sequence where the difference between consecutive terms is always the same (constant). E.g. 3, 7, 11, 15, ...
3
What is a Sequence?
+5 XP

A sequence is an ordered list of numbers following a rule. In an arithmetic sequence, you add (or subtract) the same amount each time — this fixed amount is called the common difference $d$.

Increasing sequence
3, 7, 11, 15, ... — add 4 each time, so $d = 4$ (positive)
Decreasing sequence
20, 17, 14, 11, ... — subtract 3 each time, so $d = -3$ (negative)
Finding $d$
Subtract any term from the next: $d = T_2 - T_1$. Always check with another pair to confirm.
3 7 11 15 ... +4 +4 +4 $T_1$ $T_2$ $T_3$ $T_4$ $d = T_2 - T_1 = 7 - 3 = 4$
Try It: Find $d$ for each sequence. (a) 5, 9, 13, 17, ...   (b) 30, 25, 20, 15, ... Answers: (a) $d = 4$   (b) $d = -5$
4
The nth-Term Formula
+5 XP

Instead of listing every term, we use a formula that gives any term directly. For an arithmetic sequence with first term $a$ and common difference $d$:

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

To use it: substitute $a$, $d$, and the term number $n$. Then expand and simplify to get a clean linear expression.

$T_n = a + (n-1)d$ first term common diff
$a + (n-1)d \rightarrow dn + c$

Example: Find the nth-term formula for 3, 7, 11, 15, ...

  • $a = 3$, $d = 4$
  • $T_n = 3 + (n-1) \times 4 = 3 + 4n - 4 = 4n - 1$

Check: $T_1 = 4(1) - 1 = 3$ ✓    $T_4 = 4(4) - 1 = 15$ ✓

The simplified form is always $T_n = dn + c$ — a linear expression in $n$.

Always simplify
Expand the brackets and collect like terms. The final answer should look like $dn + c$.
Always check
Substitute $n = 1$ and $n = 2$ into your formula and verify they match the first two terms.
5
Using the Formula to Find Any Term
+5 XP

Once you have the nth-term formula, you can find any term instantly — or test whether a number belongs to the sequence.

Find the 20th term of 5, 9, 13, 17, ...
$a = 5$, $d = 4$ → $T_n = 4n + 1$
$T_{20} = 4(20) + 1 = 80 + 1 = 81$
Is 45 in the sequence 3, 7, 11, 15, ... (where $T_n = 4n - 1$)?
Set $4n - 1 = 45$ → $4n = 46$ → $n = \frac{46}{4} = 11.5$ — not a whole number, so 45 is NOT in this sequence.
Integer test
Solve $T_n = \text{value}$ for $n$. If $n$ is a positive whole number, the value is in the sequence.
Non-integer means no
If $n$ comes out as a decimal or fraction, the value does not appear anywhere in the sequence.

Watch Me Solve It · Worked example

Watch Me Solve It · Finding the nth-term formula
+15 XP per step
Q
PROBLEM
Write the nth-term formula for 2, 5, 8, 11, ... and find $T_{15}$.
  1. 1
    Find the common difference
    $d = 5 - 2 = 3$
    Subtract any term from the next. Check: $8 - 5 = 3$ and $11 - 8 = 3$. Consistent — so this is arithmetic with $d = 3$.
  2. 2
    Write the formula and simplify
    $T_n = 2 + (n-1) \times 3 = 2 + 3n - 3 = 3n - 1$
    Substitute $a = 2$ and $d = 3$ into $T_n = a + (n-1)d$. Expand the bracket, then collect the constants: $2 - 3 = -1$.
  3. 3
    Find $T_{15}$
    $T_{15} = 3(15) - 1 = 45 - 1 = 44$
    Substitute $n = 15$ into the simplified formula. Multiply first, then subtract.
Answer $T_n = 3n - 1$  ·  $T_{15} = 44$
6
Common Pitfalls
heads-up
Common difference can be negative
Decreasing sequences like 20, 17, 14, 11, ... are still arithmetic. Their common difference is negative: $d = 17 - 20 = -3$.
Fix: always calculate $d = T_2 - T_1$, not the other way around. If the sequence decreases, $d$ will naturally come out negative.
$n$ starts at 1, not 0
The first term is $T_1$ (when $n = 1$). If you substitute $n = 0$ into $T_n = 3n - 1$, you get $-1$, which is NOT a term in the sequence 2, 5, 8, ...
Fix: always check your formula by substituting $n = 1$ and verifying you get the first term of the sequence.
Forgetting to simplify $a + (n-1)d$
Leaving the formula as $2 + (n-1) \times 3$ is technically correct but not fully simplified. Examiners expect the linear form $3n - 1$.
Fix: always expand the bracket — multiply $d$ through $(n-1)$ — then collect the constant terms to produce $dn + c$ form.

Quick Check · 5 questions

1
What is the term-to-term rule for 7, 12, 17, 22, 27, ...?
+10 XP
2
What is the $n$th term of 3, 7, 11, 15, 19, ...?
+10 XP
3
The $n$th term of a sequence is $5n + 2$. What is the 8th term?
+10 XP
4
The $n$th term is $3n - 1$. Is 50 in the sequence?
+10 XP
5
Which sequence has the $n$th term $2n + 3$?
+10 XP
Apply Easy 2 MARKS

Q1. Find the term-to-term rule and the next two terms of: 15, 11, 7, 3, -1, ...

Answer in your workbook.
Apply Medium 3 MARKS

Q2. Find the $n$th term of the sequence: 6, 10, 14, 18, 22, ...

Answer in your workbook.
Apply Hard 4 MARKS

Q3. The $n$th term of a sequence is $6n - 5$.

Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins

Extension Problems

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Brain Trainer

⚡
Speed Drills — Find the $n$th Term!

Set a timer for 3 minutes. Find the $n$th term for each sequence.

3, 5, 7, 9, 11, ...
4, 7, 10, 13, 16, ...
10, 8, 6, 4, 2, ...
1, 5, 9, 13, 17, ...
7, 12, 17, 22, 27, ...
20, 17, 14, 11, 8, ...
0, 3, 6, 9, 12, ...
2, 6, 10, 14, 18, ...