Question 1 Easy
What is the second difference of 2, 8, 18, 32?
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Answer: 4
- First differences 6, 10, 14.
- Those increase by 4.
- Method (1): First differences 6, 10, 14.
- Accuracy (1): Answer 4
The nth term of a quadratic sequence practice questions with answers
A second set of the nth term of a quadratic sequence questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
More questions
Question 2 Easy
The second difference of a sequence is 6. What is the coefficient of n^2?
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Answer: 3
- Halve 6.
- The square term is 3n^2.
- Method (1): Halve 6.
- Accuracy (1): Answer 3
Question 3 Medium
Find the nth term of 1, 4, 11, 22.
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Answer: 2n^2-3n+2
- Second difference 4.
- Subtract 2n^2 and fit -3n+2.
- Method (1): Second difference 4.
- Accuracy (1): Answer 2n^2-3n+2
Question 4 Medium
What is the coefficient of n^2 in the term for 1, 6, 15, 28?
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Answer: 2
- Second difference 4.
- Half of 4 is 2.
- Method (1): Second difference 4.
- Accuracy (1): Answer 2
Question 5 Hard
A sequence starts 5, 11, 21, 35. Write its nth term.
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Answer: 2n^2+3
- Second difference 4, so the square part is 2n^2.
- 2, 8, 18, 32 is 3 below each given term, so add 3.
- Method (1): Second difference 4, so the square part is 2n^2.
- Accuracy (1): Answer 2n^2+3
Question 6 Hard
Second differences of 2, 6, 12, 20 are constant. What is that constant?
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Answer: 2
- Gaps 4, 6, 8.
- The gaps rise by 2.
- Method (1): Gaps 4, 6, 8.
- Accuracy (1): Answer 2
Exam-style question
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
Exam-style question 14 marks
Find an expression for the nth term of the sequence 3, 8, 15, 24.
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Answer: n^2+2n
- Second difference 2.
- n^2 leaves 2, 4, 6, 8.
- Term n^2+2n.
- Method (1): Second difference
- Method (1): Coefficient
- Method (1): Linear part
- Accuracy (1): n^2+2n
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
What does a constant second difference tell you?
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The nth term has an n^2 part. The sequence is quadratic. The nth term has an n^2 part. The sequence is quadratic.
Check 2
How do you get the coefficient of n^2?
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Halve the second difference. Halve the second difference.
Check 3
A second difference of 2 means which square term?
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n^2. Half of 2 is 1. n^2. Half of 2 is 1.
Check 4
A second difference of 4 means which square term?
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A second difference of 4 means which square term: 2n^2. 2n^2.
Check 5
Why subtract the square part from the sequence?
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What is left is linear or constant, and you can read that next. What is left is linear or constant, and you can read that next.
Check 6
The leftovers 1, 2, 3, 4 match which expression?
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n. So you add n to the square part. n. So you add n to the square part.
Check 7
Why check n=3 after you build a rule?
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If term 3 is wrong, there is a slip somewhere in the rule, so go back and check it. If term 3 is wrong, there is a slip somewhere in the rule, so go back and check it.
Check 8
Can the nth term of 2, 8, 18, 32 be just n^2?
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No. Term 2 would be 4, but the sequence has 8. It is 2n^2. No. Term 2 would be 4, but the sequence has 8. It is 2n^2.
Check 9
Why reject n=-6 when a term equals 30?
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A term number is a positive integer. A term number is a positive integer.
Check 10
The sequence 3, 9, 19, 33 is 2n^2 plus what?
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Plus 1. Each term is one more than 2n^2. Plus 1. Each term is one more than 2n^2.
Check 11
First differences of 2, 6, 12, 20 are what?
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4, 6 and 8. 4, 6 and 8.
Check 12
Is a linear sequence allowed a second difference of 0?
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Yes. If the first differences are constant, the second difference is 0 and there is no n^2 term. Yes. If the first differences are constant, the second difference is 0 and there is no n^2 term.
Check 13
What is n^2+n when n=5?
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30. That is how you check a position. 30. That is how you check a position.
Check 14
Why not copy the second difference straight in as the coefficient?
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The second difference is twice the coefficient of n^2. The second difference is twice the coefficient of n^2.
Check 15
Name the term for 4, 10, 18, 28.
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n^2+3n. Term 1 is 1+3=4. n^2+3n. Term 1 is 1+3=4.
Keep going with The nth term of a quadratic sequence
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