🔮 Algebra Realm · Sequences

Arithmetic sequences

Arithmetic sequences focuses on how to identify sequences with a constant additive difference and use that structure. In this lesson, focus on a sequence records values in order.

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Understand Arithmetic sequences

Arithmetic sequences focuses on how to identify sequences with a constant additive difference and use that structure. In this lesson, focus on a sequence records values in order.

A sequence records values in order. An nth-term rule links any position directly to its value, so it is different from repeatedly adding a term-to-term change. For arithmetic sequences, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Arithmetic sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the arithmetic sequences representation visible until the final line.

Picture the idea

Run terms through a sequence machine and compare a step-by-step growth view with a position-to-term table. Use the model to explain one change you notice while working on arithmetic sequences.

Check as you go

Use the rule to regenerate at least two terms already shown in the sequence. Write that check beside the final arithmetic sequences answer.

Key vocabulary

termpositiondifferenceterm-to-term rulenth termarithmeticsequences

Rules and key facts

5, 8, 11, 14 has common difference 3.

  • List positions above the terms.
  • Inspect the change pattern or substitute the required position.
  • Apply the rule carefully to the requested term.
  • Check against known terms before extending the pattern. Record the check explicitly for arithmetic sequences.

Step-by-step method

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
  4. Check against known terms before extending the pattern. Record the check explicitly for arithmetic sequences.

What you need first

  • Recognise the vocabulary: term, position, difference.
  • Be able to explain the purpose of arithmetic sequences before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Tile patterns
  • Predicting repeated growth

Visual / interactive

See the idea, then move it around

Skip to Practice

Run terms through a sequence machine and compare a step-by-step growth view with a position-to-term table. Use the model to explain one change you notice while working on arithmetic sequences.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Arithmetic sequences — Find the 5th term of the sequence with nth term 3n + 2. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 5: 3 × 5 + 2 = 17. Final answer: 17. Check: substitute or compare with the original information to confirm the result fits the question.

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
Secure example

Use the normal method

Given information: Arithmetic sequences — Find the 13th term of the sequence with nth term 3n + 14. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 13: 3 × 13 + 14 = 53. Final answer: 53. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Connect the common difference to the coefficient of n when needed.

Challenge example

Stretch the idea

Given information: Arithmetic sequences — Find the 10th term of the sequence with nth term 3n + 13. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 10: 3 × 10 + 13 = 43. Final answer: 43. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Arithmetic sequences — Find the 7th term of the sequence with nth term 3n + 12. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 7: 3 × 7 + 12 = 33. Final answer: 33. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Write the common difference with its sign.

Common mistakes

  • Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering arithmetic sequences questions.
  • Starting position counting at zero without being told.

How to check your answer

Use the rule to regenerate at least two terms already shown in the sequence. Write that check beside the final arithmetic sequences answer.

Extension challenge

Create a arithmetic sequences problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Arithmetic sequences challenge

Use sequence machine controls to solve three checked arithmetic sequences rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Study cards

Core idea

Arithmetic sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the arithmetic sequences representation visible until the final line.

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Key vocabulary

term · position · difference · term-to-term rule · nth term · arithmetic · sequences

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Rules

List positions above the terms. Inspect the change pattern or substitute the required position. Apply the rule carefully to the requested term. Check against known terms before extending the pattern. Record the check explicitly for arithmetic sequences.

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Formula / fact

5, 8, 11, 14 has common difference 3.

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Foundation example

Given information: Arithmetic sequences — Find the 5th term of the sequence with nth term 3n + 2. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 5: 3 × 5 + 2 = 17. Final answer: 17. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Arithmetic sequences — Find the 13th term of the sequence with nth term 3n + 14. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 13: 3 × 13 + 14 = 53. Final answer: 53. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Arithmetic sequences — Find the 10th term of the sequence with nth term 3n + 13. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 10: 3 × 10 + 13 = 43. Final answer: 43. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Arithmetic sequences — Find the 7th term of the sequence with nth term 3n + 12. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 7: 3 × 7 + 12 = 33. Final answer: 33. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Common mistake

Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering arithmetic sequences questions.

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Exam tip

For arithmetic sequences, show the key representation before the final calculation. Use this final check: Use the rule to regenerate at least two terms already shown in the sequence.

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Real-world use

Tile patterns, Predicting repeated growth

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Checklist

I can explain arithmetic sequences, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Arithmetic sequences

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Simple explanation

Arithmetic sequences: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the arithmetic sequences representation visible until the final line.

Think of arithmetic sequences as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. List positions above the terms.
  2. Inspect the change pattern or substitute the required position.
  3. Apply the rule carefully to the requested term.
  4. Check against known terms before extending the pattern. Record the check explicitly for arithmetic sequences.

Hint 1

Start by naming the given information and the exact result required for arithmetic sequences.

Hint 2

List positions above the terms.

Full worked solution

Given information: Arithmetic sequences — Find the 5th term of the sequence with nth term 3n + 2. Method choice: use the arithmetic sequences method and show each step with the stated values. Calculation or reasoning: Substitute n = 5: 3 × 5 + 2 = 17. Final answer: 17. Check: substitute or compare with the original information to confirm the result fits the question.

Method: List positions above the terms. → Inspect the change pattern or substitute the required position. → Apply the rule carefully to the requested term. → Check against known terms before extending the pattern. Record the check explicitly for arithmetic sequences.

Common mistake warning

Using a term-to-term rule as if it were an nth-term formula. This is a key trap when answering arithmetic sequences questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain arithmetic sequences in my own words.
  • I can use these words accurately: term, position, difference.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Using a term-to-term rule as if it were an nth-term formula.
  • I can apply this check: Use the rule to regenerate at least two terms already shown in the sequence.
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