Chapter 2 · Sequences · Concept 1
2-2 Arithmetic Sequence (1)
Last lesson every sequence had its own puzzle. This one is about the friendliest kind there is: the sort where you add the same number over and over. Almost every real-world plan — a training programme, a repayment, a dose — is one of these.
Guiding question
If you keep adding the same number, what is the rule for any term at all?
Learning outcome. Calculate the general term of an arithmetic sequence, and find some specific terms in it.
Arithmetic sequences
If you get each term by adding the same number to the one before, the sequence is arithmetic, and that same number is the common difference, written d.
1, +3 4, +3 7, +3 10, +3 13, …
first term 1, common difference 3
the general term, with first term a and difference d
aₙ = a + (n − 1)d
To find d, take any term away from the one after it. Do it twice to check — if you get two different answers, the sequence is not arithmetic.
This is the single most common mistake in the whole chapter, and it is not a maths mistake — it is a counting mistake. Walk along the chain and watch the two numbers under it disagree.
you are on term 4
but you only took 3 steps
so the term is the first one, plus that many steps
a₄ = 2 + 3 × (3) = 11
the general term
aₙ = 2 + (n − 1) × (3) = 3n − 1
The first term is free.
You do not add anything to get to term 1 — you are already there. So reaching term 5 costs you four steps, not five. That missing step is the entire (n − 1).
1. Find the first term and the common difference.
2(b) 13, A, B, −2, … is arithmetic. Find d, A and B.
- 1
Count the steps between the two numbers you DO know.
13 → A → B → −2 is three arrowsThis is the counting from the chain above. Count arrows, not boxes.
3. First term 8, common difference −6. Find the general term and the 20th term.
Work with a partner. For the blanks, one of you counts the arrows out loud while the other writes. Getting the arrow count right is the whole question.
1. Find the first term and the common difference.
2. Find the common difference and the missing letters.
3. Find the general term and the 7th term.
4. Find the general term and the 10th term.
In an arithmetic sequence the 3rd and 7th terms add to 38, and the 4th term divided by the 12th term is 2/5. Find the first term and the common difference.
A student writes aₙ = a + nd instead of a + (n − 1)d. For which term number does their formula accidentally give the right answer, and why does that make the mistake so hard to spot?
Compare the two formulas term by term.
Writing a + nd.
The first term is free — you take (n − 1) steps to reach term n, not n steps. Test your formula at n = 1: it must give a.
Miscounting the arrows in a blank.
In 13, A, B, −2 there are THREE gaps, not two. Draw the arrows on the page before writing 3d = −15.
Losing a sign when you tidy.
(n − 1) × (−6) is −6n + 6, not −6n − 6. Multiply both parts of the bracket by the whole −6.
Finding d from one gap and never checking.
Work out a second gap too. If they disagree, the sequence is not arithmetic and the whole method is off.
Stopping at the general term.
If the question asks for the 20th term as well, substitute. A formula on its own is only half the marks.
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