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
What a learner can do afterwards
- 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
The lesson
You already know how to build a sequence by adding the same number again and again. Now you'll turn that pattern into one formula, called the nth-term rule. Plug in a position number, n, and the formula hands you that term, no counting needed.
Take 3, 7, 11, 15. The common difference is 4, so the rule starts with 4n. Try n = 1: 4 x 1 = 4. The real first term is 3, one less than 4. So the rule is 4n - 1. Check n = 2: 4 x 2 - 1 = 7. It matches.
There's a shortcut for that extra number. Find the zero term: subtract the common difference from the first term. For 3, 7, 11, 15, that's 3 - 4 = -1, exactly the number sitting in 4n - 1.
Once you have the rule, you can jump anywhere. For 4n - 1, the 50th term is 4 x 50 - 1 = 199, with no need to list 49 terms first. To check whether 99 belongs to this sequence, set 4n - 1 = 99. That gives 4n = 100, so n = 25. Since 25 is a whole number, 99 is the 25th term.
An nth-term rule is (common difference) x n + (zero term); use it to jump straight to any term or to test whether a number belongs to the sequence.
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Where it sits
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Where this leads
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.