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
1 · Read
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.
More to read
13.1 Sequences and Their Notations
A sequence is a list of numbers in order, and the position matters as much as the value. The 1st term, 2nd term, 3rd term each have an address n, and the nth-term rule is the machine that turns the address into the value. Listing terms is slow; the rule jumps straight to any position.
Spotting the gap is the first skill. In 3, 7, 11, 15 the gap is 4 every time, which means the rule involves 4n. Adjusting 4n to hit the actual terms gives 4n - 1. One rule replaces the whole list.
13.2 Arithmetic Sequences
Arithmetic sequences add a constant gap each step, and their rules all look like an plus b. The multiplier a is the gap, and b shifts the line of multiples onto the terms. For 2, 5, 8, 11 the gap is 3, and shifting 3n down by 1 lands on every term: 3n - 1.
The rule earns its keep on far-out terms. Finding the 50th term by listing would take all lesson; substituting n = 50 takes seconds. Testing membership reverses the flow: set the rule equal to the candidate and check whether n comes out whole.
13.3 Geometric Sequences
Not every sequence adds: geometric sequences multiply by a constant ratio instead. Doubling each time gives 2, 4, 8, 16, where the rule uses powers rather than multiples. Meeting both families trains the eye to ask which operation links the terms.
The habit transfers back to arithmetic work. Checking a rule against a fresh term, like testing n = 6 in 4n + 1 to get 25, confirms the pattern before trusting it far out. Whether adding or multiplying, the rule must earn trust term by term.
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 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.