🔮 Algebra Realm · Sequences

nth term

nth term focuses on how to derive a direct formula for any term in an arithmetic sequence. In this lesson, focus on a sequence records values in order.

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Understand nth term

nth term focuses on how to derive a direct formula for any term in an arithmetic sequence. 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 nth term, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

nth term: A sequence records values in order. Use the rule to regenerate at least two terms already shown in the sequence. Keep the nth term 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 nth term.

Check as you go

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

Key vocabulary

Rules and key facts

4, 7, 10, 13 has nth term 3n + 1.

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 nth term.

Step-by-step 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 nth term.

What you need first

Recognise the vocabulary: term, position, difference.
Be able to explain the purpose of nth term 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 nth term.

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

Responsive · validated · topic linked

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: nth term — Find the 5th term of the sequence with nth term 3n + 2. Method choice: use the nth term 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.

List positions above the terms.
Inspect the change pattern or substitute the required position.
Apply the rule carefully to the requested term.

Secure example

Use the normal method

Given information: nth term — Find the 13th term of the sequence with nth term 3n + 14. Method choice: use the nth term 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: Add or subtract the fixed adjustment and test several positions.

Challenge example

Stretch the idea

Given information: nth term — Find the 10th term of the sequence with nth term 3n + 13. Method choice: use the nth term 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: nth term — Find the 7th term of the sequence with nth term 3n + 12. Method choice: use the nth term 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.

Common mistakes

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

Extension challenge

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

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nth term challenge

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

Press Start Game to enter a topic-specific maths arena.

Boss challenge

Challenge nth term Arcane Boss

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nth term Arcane Boss

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Study cards

Core idea

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

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Key vocabulary

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

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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 nth term.

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Formula / fact

4, 7, 10, 13 has nth term 3n + 1.

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Foundation example

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Secure example

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Challenge example

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Exam-style example

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Common mistake

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

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Exam tip

For nth term, 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 nth term, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Simple explanation

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

Think of nth term 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

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 nth term.

Hint 1

Start by naming the given information and the exact result required for nth term.

Hint 2

List positions above the terms.

Full worked solution

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 nth term.

Common mistake warning

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

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

I can explain nth term 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.

🥉 Bronze Badge Foundation level completed

🥈 Silver Badge Apprentice level completed

🥇 Gold Badge Skilled level completed

💎 Platinum Badge Mastery test passed

🐉 Legendary Badge Boss battle defeated