GCSE · Maths · AQA · Spec 8300 · Foundation

nth term of linear sequences

5, 8, 11, 14, … What's the 100th term? You could keep adding 3 — or find one short rule that jumps straight there. That rule is the nth term.

5, 8, 11, 14, … is the 3 times table in disguise

Position n: 2. Term: 8

Drag the point to n = 1, 2, 3 and 4, then all the way back to n = 0.

Red line: the 3 times table, 3n: Stop the point on n = 1, 2, 3 and 4 and read the blue terms: 5, 8, 11, 14. The red line gives 3, 6, 9, 12 at the same positions. Both climb 3 for every step along, and that's why the rule has 3n in it. Now compare them: the blue is 2 higher every time (5 − 3, 8 − 6, 11 − 9, 14 − 12). So the nth term is 3n + 2. Only whole-number positions are terms. At n = 0 the blue line reads 2, which is the term that would come just before 5. That's where the + 2 comes from.

Exam line: The number in front of n is the common difference. The number on the end is the gap between the sequence and that times table, which is also the value at n = 0, one step back from the first term.

Your turn to think

Which rule would you write?

Here's a new sequence: 9, 14, 19, 24, … It goes up by 5 each time.

Which expression for the nth term is closest to what you'd write?

n + 5, because you add 5 each time.5n + 9, because it goes up in 5s and starts at 9.5n, because it goes up in 5s, so it's the 5 times table.5n + 4, because it's the 5 times table with 4 added on.

How sure are you?

I'm sureI think soI'm guessing

n + 5, because you add 5 each time.

This is a common misconception

"Add 5" is the term-to-term rule: it gets you from one term to the next. The nth term has to turn a position into a term. Try it: n = 1 gives 1 + 5 = 6, not 9, and n = 2 gives 7, not 14. The expression n + 5 only goes up by 1 each step. To go up by 5 each step you need 5n.

5n + 9, because it goes up in 5s and starts at 9.

This is a common misconception

The 5n is right, but the + 9 isn't. At n = 1, 5n already gives 5, so 5n + 9 gives 14. That's the second term, not the first. The + 9 lands on top of the 5 the times table has already given you. The number on the end is the gap from the times table: 9 − 5 = 4.

5n, because it goes up in 5s, so it's the 5 times table.

This is a common misconception

Half right. The terms do climb in 5s, so the 5 times table is underneath. But 5n gives 5, 10, 15, 20, and the sequence is 9, 14, 19, 24. Every term is 4 more than that, and that fixed gap has to go into the rule as well.

5n + 4, because it's the 5 times table with 4 added on.

This matches the evidence

Yes. The difference of 5 gives 5n: 5, 10, 15, 20. Every term is 4 more than that, so add 4. Check it: n = 1 gives 5 + 4 = 9 ✓ and n = 2 gives 10 + 4 = 14 ✓. Notice that 4 is exactly what you'd get by stepping back one place from 9.

The difference goes in front of n. The number on the end is whatever turns that times table into your sequence. It isn't the first term, and it isn't the difference.

The method, written out

Problem

Find an expression for the nth term of 7, 11, 15, 19, 23, …

Now you choose the steps

A sequence that goes down

Find an expression for the nth term of 20, 17, 14, 11, 8, …

20, 17, 14, 11, 8, … Each term is 3 less than the one before, so the common difference is −3.
missing step

Which line is step 2?

The number in front of n is 3, so compare the sequence with 3n.The number in front of n is −3, so compare the sequence with −3n.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every linear sequence is a times table in disguise. Work out which times table, then how far it has been shifted, and you have a rule that jumps straight to any term.

What you need to know

A linear sequence goes up or down by the same amount every time. That amount is the common difference.
The nth term is a rule that takes a position number n and gives you the term in that position.
For a linear sequence the rule has the form (common difference) × n, then add or subtract a fixed number.
Check any nth term you write by substituting a position whose term you already know.

The big picture

A linear sequence goes up or down by the same amount every time: the common difference. Its nth term is a rule that turns a position n into the term at that position. The common difference goes in front of n. Then compare the sequence with that times table: the gap between them is the number you add or subtract. Finally, check your expression by putting in a position you know.

Key points

1The number in front of n is the common difference, sign included: terms that go down by 3 give −3n.

2The number on the end is the gap between the sequence and the matching times table. It is also the value at n = 0: the term that would come just before the first one.

3When the sequence sits below its times table, the number on the end is negative, as in 6n − 5.

423 − 3n and −3n + 23 are the same expression written in a different order.

5Once you have the nth term, you can find any term by substituting its position.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

Work out an expression for the nth term of 1, 7, 13, 19, 25, … Then use it to find the 50th term.

⚠ Watch out

Writing the term-to-term rule as the nth term. For 2, 5, 8, 11, … "add 3" gets you from one term to the next, but n + 3 gives 4, 5, 6, 7 — it only climbs by 1. Climbing by 3 needs 3n: here, 3n − 1.

🧠

Memory hook

Which times table? How far off it? Does n = 1 work?

✓

Check yourself

Find the nth term of 2, 9, 16, 23, … Then check that your rule gives 16 when n = 3.

Flashcards

(7)

What does the nth term of a sequence let you do?

Work out any term straight from its position n, without listing all the terms before it.

What makes a sequence linear?

It goes up or down by the same amount every time. That amount is the common difference.

In the nth term of a linear sequence, what number goes in front of n?

The common difference, including its sign.

How do you find the number on the end of the nth term?

Compare the sequence with the times table of the difference. The gap between them is the number you add or subtract.

Put n = 0 into an nth term such as 3n + 2. What do you get, and what does it mean?

Just the number on the end (here 2). It is the term that would come just before the first term.

A sequence goes DOWN by 4 each time. What goes in front of n?

−4. A negative difference gives a negative number in front of n.

What are the four steps for finding the nth term of a linear sequence?

1. Find the common difference d and write dn. 2. Compare with the d times table. 3. Add or subtract the gap. 4. Check by substituting a known position.

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.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

5, 8, 11, 14, … is the 3 times table in disguise

Which rule would you write?

A sequence that goes down

More AQA GCSE Maths topics