GCSE · Maths · AQA · Spec 8300 · Higher

nth term of quadratic sequences (Higher)

Some sequences speed up as they go, and one clever subtraction splits their rule into two easy pieces.

Lift the 5th term and watch the gaps

01.534.560306090120Position nTerm2Second differencelift me
Coefficient a 1

Coefficient a: 1. Second difference: 2

The line joins the terms of an² + 2n + 3, from n = 1 to n = 5. Each straight piece climbs by one first difference. At a = 1 the terms are 6, 11, 18, 27, 38: the gaps are 5, 7, 9, 11, and each gap is 2 more than the last. At a = 2 the terms are 7, 15, 27, 43, 63: the gaps are 8, 12, 16, 20, each 4 more than the last.

Watch out: The second difference is 2a, not a. If you see a second difference of 6, that's 3n², not 6n².

Linear, quadratic or neither?

Work out the first differences of each sequence (and the second differences if you need them), then put it in the right group.

Still to sort

Linear (0)

The first differences are all the same.

Where the line is: If the first differences are already equal, stop there: it's linear. You only need second differences when the first ones keep changing.

Quadratic (0)

The first differences change, but the second differences are all the same.

Where the line is: Growing faster and faster isn't enough. The second differences have to be exactly equal.

Neither (0)

Even the second differences keep changing.

7 of 7 still to sort.

Before you can find a rule, you need to know what kind of sequence you've got. Work out the differences, then sort.

Watch it done once, slowly

Problem

Find the nth term of 4, 13, 26, 43, 64, …

Spot the slip

Find the nth term of 2, 9, 18, 29, 42, …

A student's answer — which line goes wrong?

Your turn: fill in the missing steps

Find the nth term of 5, 12, 25, 44, 69, …

  1. First differences: 7, 13, 19, 25. Second differences: 6, 6, 6. They're equal, so it's quadratic.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find the n² part, peel it off, and a plain linear sequence is waiting underneath.

What you need to know

  • A quadratic sequence has no common difference, but its second differences are all equal.
  • The nth term of a quadratic sequence has the form an² + bn + c.
  • a is half the second difference, because n² on its own (1, 4, 9, 16, …) has a second difference of 2.
  • Take an² away from each term. The sequence left over is linear, and its nth term is bn + c.

The big picture

In a quadratic sequence the gaps between terms keep changing, but the gaps between those gaps (the second differences) are all equal. Its nth term has the form an² + bn + c. Because n² on its own has a second difference of 2, a is half the second difference. Take an² away from every term and a linear sequence is left, and its nth term gives you bn + c.

Key points

1Equal second differences mean the sequence is quadratic.
2Second difference = 2a, so a = second difference ÷ 2.
3a can be a fraction (second difference 1 gives ½n²) or negative (−4 gives −2n²).
4Terms minus an² leaves a linear sequence: bn + c.
5Check your final rule by substituting a value of n and comparing with that term.

Worked example

Problem

Find the nth term of the sequence 12, 17, 18, 15, 8, …

⚠ Watch out

Using the second difference itself as a. A second difference of 6 means 3n², not 6n², so always halve it.

🧠

Memory hook

Halve, peel, line up: halve the second difference to get a, peel an² off every term, then line up the linear rule for what's left.

✓

Check yourself

When you take an² away from every term of a quadratic sequence, why is the sequence that's left always linear?

Flashcards

(11)
How can you tell a sequence is quadratic?
The first differences change, but the second differences are all equal.
What is the general form of the nth term of a quadratic sequence?
an² + bn + c
What is the second difference of n² (1, 4, 9, 16, 25, …)?
2. The first differences are 3, 5, 7, 9, and they go up by 2 each time.
The second differences are all 10. What is a?
5. a is half the second difference, so the n² part is 5n².
The second differences are all 1. What is a?
½, so the n² part is ½n². a doesn't have to be a whole number.
The second differences are all −6. What is a?
−3, so the n² part is −3n². Taking −3n² away from each term means adding 3n².
You've found a. What's your next step?
Take an² away from every term of the sequence.
What does the leftover sequence give you?
bn + c, the rest of the nth term. Find it the same way as any linear nth term.
Your leftover sequence isn't linear. What has gone wrong?
Your value of a is wrong. Most often you forgot to halve the second difference.
Do b and c change the second difference?
No. Changing b or c moves the terms, but only the n² part decides how fast the gaps grow.
How do you check your nth term?
Substitute a value of n, such as n = 3, and see if you get that term of the sequence.

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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