Sequences and the nth term, linear and quadratic
Find the nth term using staircases. Worked examples. Exam-style questions with answers.
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Download the PDF (624 KB)How can you find the 100th number in a sequence without writing out the first 99?
Sequences feel hard because they hide patterns. Think of them as staircases. Each step shows how numbers change.
The picture to keep in your head: A staircase
A linear sequence is a staircase with equal steps: 5, 8, 11, 14 goes up 3 every step, so the nth term is 3n plus where the staircase starts. Step back one stair from the first term to find the start (the zeroth term, 2), so the nth term is 3n + 2. A quadratic sequence is a staircase whose steps grow steadily: the steps themselves go up by the same amount each time (the second difference), and half of that is the number in front of n2.
The method, every time
- Find the step size by subtracting any two terms.
- Step back one from the first term to find the zeroth term.
- Write the nth term as step size times n plus the zeroth term.
- Check by plugging in n = 1 and n = 2 to see if you get the right numbers.
- For quadratics, find half the second difference and use it as the n squared term.
Words you'll see
- nth term
- The rule to find any number in a sequence.
- step size
- How much each term increases by.
- zeroth term
- The number before the first term in a staircase.
- second difference
- The change in step size for quadratic sequences.
- quadratic sequence
- A sequence where steps grow by a steady amount.
Step size times n, plus the zeroth term
A linear sequence is like a staircase with equal steps. The step size is the common difference, and the zeroth term is where the staircase starts. The nth term is step size times n plus the zeroth term. Find the step size first, then subtract it from the first term to get the zeroth term.
Watch out
- Writing n + 3 instead of 3n. Multiply step size by n, not add it
- Using the first term instead of the zeroth term. Subtract step size from first term to find zeroth term
Find any term, or test a number
A staircase helps you find any step by counting how many steps you climb. To test if a number is in the sequence, imagine climbing to that step and seeing if it lands exactly on a rung. The rule is: substitute n into the nth term to find a term, or set the nth term equal to the number and solve for n.
Watch out
- Saying yes when n isn't a whole number. Only whole-number n means the number is in the sequence
- Substituting the number for n instead of solving. Set the nth term equal to the number and solve for n
Half the second difference
A quadratic sequence is like a staircase where the steps themselves grow by the same amount each time. The second difference shows how fast the steps are growing. The coefficient of n² is half the second difference. Once you know that, subtract the n² part from each term and find the remaining linear part. Add them together to get the full nth term.
Watch out
- Using the whole second difference as the n² coefficient. Divide the second difference by 2 to get the n² coefficient
- Subtracting the n² part from the wrong term. Subtract n² from each term in order, starting from the first term
Geometric and Fibonacci-type
A geometric sequence is like a staircase that grows by multiplying each step. A Fibonacci-type sequence is like a staircase where each step is the sum of the two before. To find the next term, check if you multiply or add the previous terms. The rule is: geometric sequences multiply by the same number each time, Fibonacci-type sequences add the two previous terms.
Watch out
- Looking only at differences when the rule multiplies. Check if the ratio between terms is constant, not the difference.
- Adding the step to the wrong term. Add the previous two terms, not just one.
21 exam-style sequences and the nth term questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Linear nth term
- 1Find the nth term of 3, 6, 9, 12, 15, …[2 marks]
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3n
Step 3, zeroth term 0. - 2Find the nth term of 5, 8, 11, 14, 17, …[2 marks]
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3n + 2
Step 3, zeroth term 5 - 3 = 2. - 3Find the nth term of 10, 7, 4, 1, -2, …[2 marks]
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-3n + 13
Step -3, zeroth term 10 + 3 = 13. - 4Find the nth term of 2.5, 5, 7.5, 10, 12.5, …[2 marks]
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2.5n
Step 2.5, zeroth term 0. - 5Find the nth term of -4, -1, 2, 5, 8, …[2 marks]
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3n - 7
Step 3, zeroth term -4 - 3 = -7.
B · Use it
- 6Find the 50th term of 2.5, 5, 7.5, 10, …[2 marks]
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125
nth term 2.5n, so 2.5 × 50. - 7Is 200 a term of -4, -1, 2, 5, 8, …? Give a reason.[2 marks]
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Yes: it is the 69th term
nth term 3n - 7 = 200 gives n = 69, a whole number. - 8Find the first term greater than 100 in 4, 11, 18, 25, …[2 marks]
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102
nth term 7n - 3 > 100 gives n > 14.7, so n = 15: 7 × 15 - 3 = 102. - 9Find the 50th term of -4, -1, 2, 5, 8, …[2 marks]
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143
3 × 50 - 7 = 143.
C · Quadratic (Higher)
- 10Find the nth term of 2, 8, 18, 32, 50, …[3 marks]
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2n²
Second difference 4, so 2n²; nothing is left over. - 11Find the nth term of 4, 11, 22, 37, 56, …[3 marks]
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2n² + n + 1
Second difference 4, so 2n². Left over: 2, 3, 4, 5, 6 = n + 1. - 12Find the nth term of 4, 13, 28, 49, 76, …[3 marks]
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3n² + 1
Second difference 6, so 3n². Left over: 1, 1, 1, … = 1. - 13Find the nth term of 3, 6, 11, 18, 27, …[3 marks]
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n² + 2
Second difference 2, so n². Left over: 2 each time. - 14Find the nth term of 5, 10, 17, 26, 37, …[3 marks]
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n² + 2n + 2
Second difference 2, so n². Left over: 4, 6, 8, … = 2n + 2.
D · Other sequences
- 15Find the term-to-term rule for the sequence: 2, 6, 10, 14, 18.[3 marks]
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Add 4
Look at the difference between terms. 6 - 2 = 4, 10 - 6 = 4, etc. So each term is 4 more than the previous. Rule: add 4. - 16Find the missing terms in the Fibonacci-type sequence: 3, 5, ?, ?, ?.[3 marks]
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8, 13, 21
Each term is the sum of the two previous terms. 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21. Missing terms: 8, 13, 21. - 17Find the next term in the sequence: √2, 2√2, 3√2, 4√2, ?.[3 marks]
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5√2
Each term increases by √2. 4√2 + √2 = 5√2. Next term is 5√2. - 18Find the next term in the sequence: 1/2, 1/4, 1/8, 1/16, ?.[3 marks]
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1/32
Each term is half of the previous. 1/16 ÷ 2 = 1/32. Next term is 1/32.
E · Patterns
- 19A matchstick pattern has 5 sticks in pattern 1 and increases by 3 sticks each time. Find the nth term and the number of sticks in pattern 20.[3 marks]
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nth term: 3n + 2, pattern 20: 62
Step 1: First term is 5. Step 2: Difference is 3. Step 3: nth term is 3n + 2. Step 4: Pattern 20 is 3×20 + 2 = 62. - 20A matchstick pattern has 8 sticks in pattern 1 and increases by 4 sticks each time. Find the nth term and the number of sticks in pattern 20.[3 marks]
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nth term: 4n + 4, pattern 20: 84
Step 1: First term is 8. Step 2: Difference is 4. Step 3: nth term is 4n + 4. Step 4: Pattern 20 is 4×20 + 4 = 84. - 21A matchstick pattern has 12 sticks in pattern 1 and increases by 5 sticks each time. Find the nth term and the number of sticks in pattern 20.[3 marks]
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nth term: 5n + 7, pattern 20: 107
Step 1: First term is 12. Step 2: Difference is 5. Step 3: nth term is 5n + 7. Step 4: Pattern 20 is 5×20 + 7 = 107.
Sequences and the nth term exam checklist
- Step size times n
- Zeroth term = first term minus the step
- Check with n = 1 and 2
- Quadratic: half the second difference
- Whole-number n means the number is in the sequence
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