Nth-Term Rules
Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence
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Teaching Table
Set out your lesson
Find an arithmetic sequence's nth-term rule, calculate a distant term, and test membership with a valid position.
First use Generating Sequences. Ask the learner to explain the difference between a position and a value. Ask them to substitute n = 2 into 3n + 1. Expect 7. If substitution is insecure, return to Substituting into Formulae. Solving Linear Equations is an absent soft prerequisite; the limited undoing method required here is taught below, not assumed.
Check readiness directly. Prerequisite availability is not attainment.
Start with the readiness check. Use Show, then Guided, then a fresh Try. These linked lessons may need several short sessions; do not impose a speed target. Read the active sheet on screen and answer on plain paper or by typing. No printer, cutting, purchase or internet is needed. The adult may read the exact prompt. Record any hints, scribing or calculation help. Oral explanations are allowed but do not show independent reading or handwriting. Use as much space and time as needed; pause or split the lesson. Keep this adult guide and all keys private. Share only the active learner sheet. Before Try, put Show, Guided work and notes aside. Keep the fresh Try givens available. Give no answer or target method during the first attempt. If help is needed, give it and mark the attempt supported. Review the key after the attempt. Record the actual response and help; do not invent success or infer mastery from one attempt. These are fictional paper or oral mathematics tasks. No personal information, public sharing, purchase, physical experiment or real measurement is required.
Record the actual response and assistance. These tasks do not establish mastery.
Use the complete materials on screen or print only the current sheet. Keep adult guidance out of learner view. Retain an earlier model only when the current step supplies it as context.
Teaching limits and safe use
These are finite task opportunities, not proof of mastery. Record what was attempted and what support was used.
Only increasing arithmetic sequences with positive integer differences are used. The membership method does not teach the full topic of solving linear equations.
Record the actual response and assistance. These tasks do not establish mastery.
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Teaching approaches
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1 approach
Find an arithmetic sequence's nth-term rule, calculate a distant term, and test membership with a valid position.
Find an arithmetic sequence's nth-term rule, calculate a distant term, and test membership with a valid position.
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Learning resources
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Check understanding
- Derive the nth-term rule for an arithmetic sequence such as 3, 7, 11, 15, … as 4n − 1
- Use an nth-term formula to find the 50th or 100th term without listing all preceding terms
- Determine whether a given number (e.g., 99) is a term in a specified arithmetic sequence
“If your child is given the sequence 4, 7, 10, 13…, can they find a formula for the nth term — and use it to check whether, say, 100 is in the sequence?”
Curriculum record
- Type
- Procedural
- Subject
- Mathematics
- Domain
- Algebra
- Age range
- Ages 12–14
Standards