KS3 · Maths

Arithmetic sequences and finding the nth term

What is the 100th number in 5, 9, 13, 17, 21…? Don't write them all out: this list is just a times table, nudged along.

Maths · Sequences

Slide the times table

Two lists, side by side. The first is the start of the 4 times table (4×1, 4×2, 4×3). The second is the start of a sequence: 5, 9, 13. Drag every tag to its value on the line, then check — and watch what happens at each pair: 4 and 5, then 8 and 9, then 12 and 13.

Spot the arithmetic ones

One test, several look-alikes

Tick every property that is true for each sequence, then check. Only the first column is the test for 'arithmetic' — the other three are separate questions.

13, 11, 9, 7
3/5, 2/5, 1/5, 0
2, 4, 8, 16
5, 10, 15, 25
−5, −3, −1, 1, 3

Predict, then check

Count the steps between the known terms, not the terms.

12, _, 18 is an arithmetic sequence. What is the missing term?

What does the rule really say?

Reading 3n + 2

A sequence has the rule 3n + 2, where n is the term number.

Which of these is closest to what you think the rule means right now?
How sure are you?

Your turn to drive

Find the nth term of 8, 14, 20, 26, 32

Find the nth term of 8, 14, 20, 26, 32.

  1. Find the jumps: 14 − 8 = 6, 20 − 14 = 6, 26 − 20 = 6. The same every time, so this is arithmetic with common difference 6.
  2. missing step
Which line is step 2?

Catch the slip

Where does this answer go wrong?

A student finds the nth term of 100, 90, 80, 70, 60. Pick the line where the working goes wrong.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Spot the equal jumps, then write the rule that finds any term.

What you need to know

  • An arithmetic sequence goes up or down by the same step every time. That step is the common difference, and it can be negative or a fraction.
  • n is the term number: n = 1 is the 1st term, n = 10 is the 10th. It is a position, never a value in the sequence.
  • In an + b, a is the common difference and b is the translation from the a times table.
  • To find the nth term: find the common difference, write the matching times table underneath, see how far each term is from it, then check with n = 1.

The big picture

In an arithmetic sequence every term is the same step away from the one before: the common difference. The nth term an + b packs that into a rule. a is the common difference, and b is the translation from the a times table.

Key points

1Arithmetic means a constant first difference. If the differences themselves change, even by the same amount each time, it is not arithmetic.
2A common difference alone doesn't pin a sequence down. You also need the first term.
3Substituting n = 1, 2, 3… into a rule generates the terms: 3n + 2 gives 5, 8, 11, 14, 17.
4A positive coefficient of n gives an increasing sequence; a negative coefficient gives a decreasing one.
5Use a rule both ways: substitute a term number to get a term, or approximate and count on to find a term's number.

Worked example

Problem

(a) Find the 100th term of 4n + 5. (b) For 5n + 1 the 5th term is 26. Is the 10th term 52? (c) Which term of the sequence with nth term 2n − 9 is 399?

⚠ Watch out

Reading the number on its own as the step. In n + 3 the step is 1, not 3: the rule gives 4, 5, 6, 7, 8, and the + 3 only moves the whole run along. The step is the number multiplying n.

🧠

Memory hook

Times table first, then slide: the number with n is the step, the number on its own is the slide.

✓

Check yourself

Without writing out many terms: 7, 11, 15, 19 is an arithmetic sequence. What is its common difference, and what is its nth term? Then say how you would check your rule in one step.

Flashcards

(15)
What makes a sequence arithmetic?
The difference between successive terms is the same every time. The terms go up or down by the same step.
Is 2, 4, 8, 16 arithmetic? What about 5, 10, 15, 25?
Neither. The steps are +2, +4, +8 and +5, +5, +10. They are not constant.
Can an arithmetic sequence go down, use fractions or cross zero?
Yes to all three. 13, 11, 9, 7 adds −2, 3/5, 2/5, 1/5, 0 adds −1/5, and −5, −3, −1, 1, 3 has negative and positive terms.
Is b, b + c, b + 2c, b + 3c arithmetic? What about x, x + b, x + 3b, x + 6b?
The first is: first term b, common difference c. The second is not, because its steps are b, 2b, 3b.
Is a common difference enough to describe a sequence?
No. 10, 20, 30 and 55, 65, 75 both have common difference 10. You also need the first term.
How do you find the missing term in 33, _, _, 21 if it is arithmetic?
Three subtractions cover the drop of 12, so each step is 12 ÷ 3 = 4. Subtract 4 each time.
An arithmetic sequence has 3rd term 15 and goes up by 10. What is the 1st term?
Step back twice: 15 − 10 − 10 = −5.
What does n stand for in an nth-term rule?
The term number, which is the position. n = 10 means the 10th term, not that 10 is in the sequence.
What are the first five terms of 3n + 2?
5, 8, 11, 14, 17. Substitute n = 1, 2, 3, 4, 5. It does not start at 3 and add 2.
In an + b, what do a and b tell you?
a is the common difference. b is the translation: how far each term is from the matching term of the a times table.
How do you find the nth term of 5, 9, 13, 17, 21?
Common difference 4, so compare with 4, 8, 12, 16, 20. Each term is 1 more, so the nth term is 4n + 1.
How can you check an nth-term rule quickly?
Substitute n = 1 and see if you get the first term. −5, −3, −1 is 2n − 7, not 2n − 5, because 2n − 5 gives −3.
What is the nth term of 100, 90, 80, 70, 60?
110 − 10n. The common difference is −10 and the translation from −10n is +110. A negative coefficient means a decreasing sequence.
Can you double the 5th term to get the 10th term of 5n + 1?
No. The 5th term is 26 but the 10th is 5 × 10 + 1 = 51, not 52. The fives double, the 1 does not.
How do you find which term of 2n − 9 is 399?
Approximate, then count on. The 200th term is 391, and counting on in twos gives 399 as the 204th term.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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