When a sequence goes up or down by the same amount each time, one short formula gives any term you like: the 10th, the 100th, or the first one past 1000.
If the terms go up by \(d\) each time, the nth term is \(dn + c\). Find \(d\) from the common difference, then \(c\) from the first term: \(c = \text{first term} - d\).
Fig. 1: one square needs 4 sticks and each extra square adds 3 (blue), so pattern n uses 3n + 1 sticks
Section 01
Spotting the rule.
In the lab, set a first term and a common difference and watch the terms build up as bars. Change the difference to a negative number and see the sequence go downhill. Then read off the nth term the lab gives and check it against the first few terms.
Term-to-term rule: how to get from one term to the next, e.g. "add 4".
Linear sequence: the difference between terms is constant, e.g. 7, 11, 15, 19 (difference 4).
Not linear: 1, 3, 6, 10, 15 has differences 2, 3, 4, 5, which change.
Notation: \(T_n\) means the nth term, so \(T_1\) is the first term and \(T_{10}\) the tenth.
Worked example Continue a sequence
Write down the next two terms and the term-to-term rule: \(7, 11, 15, 19, \ldots\)
Differences: \(11 - 7 = 4\), \(15 - 11 = 4\), \(19 - 15 = 4\). Constant, so the rule is add 4.
\(19 + 4 = 23\), then \(23 + 4 = 27\).
Answer: 23, 27; rule: add 4
Worked example Changing differences
Write down the next two terms: \(1, 3, 6, 10, 15, \ldots\)
Differences: 2, 3, 4, 5. Each difference is one more than the last.
Next differences are 6 and 7: \(15 + 6 = 21\), \(21 + 7 = 28\).
Answer: 21, 28 (these are the triangle numbers)
Examiner trap
A term-to-term rule (add 4) is not an nth term. To find the 100th term with it you would need all 99 terms before. The nth term formula jumps straight there.
Section 02
The nth term formula.
In a linear sequence the common difference \(d\) is the number in front of \(n\). Compare the sequence with the times table of \(d\) to find what to add or subtract.
\(n\)
1
2
3
4
\(3n\)
3
6
9
12
Sequence
5
8
11
14
Difference
every term is 2 more than \(3n\), so \(3n + 2\)
Worked example Increasing sequence
Find the nth term of \(5, 8, 11, 14, \ldots\)
The common difference is 3, so the formula starts \(3n\).
When \(n = 1\), \(3n = 3\), but the first term is 5. Add 2.
Check \(n = 4\): \(3(4) + 2 = 14\). Correct.
Answer: \(3n + 2\)
Worked example Decreasing sequence
Find the nth term of \(20, 17, 14, 11, \ldots\)
The terms go DOWN by 3, so \(d = -3\) and the formula starts \(-3n\).
When \(n = 1\), \(-3n = -3\), but the first term is 20. Add 23.
Check \(n = 2\): \(23 - 6 = 17\). Correct.
Answer: \(23 - 3n\)
Your turn
Find the nth term of \(9, 16, 23, 30, \ldots\) and use it to find the 50th term.
\(d = 7\); \(7(1) = 7\) and the first term is 9, so \(7n + 2\). 50th term: \(7(50) + 2 = 352\).
Examiner trap
For 5, 8, 11, 14 do not write \(n + 3\) (adding the difference) or \(3n + 5\) (adding the first term). Always check your formula with \(n = 1\) AND \(n = 2\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
What is the next term of \(4, 9, 14, 19, \ldots\)?
Quick check+10 XP
Find the nth term of \(6, 10, 14, 18, \ldots\)
Quick check+10 XP
The nth term of a sequence is \(5n - 3\). What is the 10th term?
Quick check+10 XP
The first term of a sequence is 40 and the term-to-term rule is subtract 6. What is the 4th term?
Section 03
Using the nth term.
Once you have the formula, set it equal to a number (or put it in an inequality) and solve for \(n\). The position \(n\) must be a positive whole number.
Worked example Is it in the sequence?
Is 150 a term of the sequence with nth term \(3n + 2\)? Is 152?
\(3n + 2 = 150\) gives \(3n = 148\), so \(n = 49.33...\). Not a whole number, so 150 is NOT a term.
\(3n + 2 = 152\) gives \(3n = 150\), so \(n = 50\). 152 is the 50th term.
Answer: 150 is not a term; 152 is the 50th term.
Worked example First term over 100
Find the first term of the sequence \(7n - 4\) that is greater than 100.
\(T_{15} = 7(15) - 4 = 101\). Check \(T_{14} = 94\), which is not over 100.
Answer: 101
Worked example Two sequences (Extended)
Sequence P has nth term \(4n - 1\). Sequence Q is \(6, 14, 22, 30, \ldots\) Find the nth term of Q.
P is \(3, 7, 11, 15, \ldots\)
Compare: \(6 = 2 \times 3\), \(14 = 2 \times 7\), \(22 = 2 \times 11\). Each term of Q is double the term of P.
So Q is \(2(4n - 1) = 8n - 2\). Check: difference 8, and \(8 - 2 = 6\).
Answer: \(8n - 2\)
Examiner trap
When the answer to \(n\) is a decimal, do not round it and claim the number is a term. A decimal \(n\) means the number is NOT in the sequence.
Section 04
Sequences from patterns.
Count the objects in the first few patterns, find the nth term, then use it. Fig. 1 is a typical one: squares made from matchsticks.
Worked example Matchstick squares
Pattern \(n\) is a row of \(n\) squares made from matchsticks (Fig. 1). (a) How many sticks are in pattern 25? (b) What is the largest pattern that can be made with 200 sticks?
Sticks: 4, 7, 10, ... Difference 3, first term 4, so the nth term is \(3n + 1\).
(a) \(3(25) + 1 = 76\).
(b) \(3n + 1 \le 200\), so \(3n \le 199\) and \(n \le 66.33\).
Pattern 67 would need 202 sticks, too many. So round DOWN.
Answer: (a) 76 sticks (b) pattern 66
Examiner trap
A pattern number must be a whole number. Rounding 66.33 is not about 'nearest': round in the direction that keeps you within the 200 sticks.
Stage 2 · Exam
Exam-style questions
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Exam style+20 XP
Find the nth term of \(23, 19, 15, 11, \ldots\)
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Which of these numbers is a term of the sequence with nth term \(6n - 4\)?
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Pattern \(n\) is a row of \(n\) matchstick squares, using \(3n + 1\) sticks. What is the largest pattern that can be made with 200 sticks?
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Find the first term of the sequence with nth term \(7n - 4\) that is greater than 100.
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