Same Math › Algebra 1 · grades 8–10 › Arithmetic and geometric sequences
How do arithmetic and geometric sequences work?
A sequence is a list of numbers that follows a rule. In an arithmetic sequence you add the same number each time (the common difference, d): 3, 7, 11, 15. In a geometric sequence you multiply by the same number (the common ratio, r): 2, 6, 18, 54. Kids find d or r, extend the list, and write a formula for the nth term so they can jump to the 50th without listing.
- Arithmetic: subtract neighbors to find d. aₙ = a₁ + d(n − 1).
- Geometric: divide neighbors to find r. aₙ = a₁ · rⁿ⁻¹.
- Check: does the rule work for every pair?
Why schools teach it
Linear and exponential in disguise
Algebra 1 treats sequences as functions whose inputs are 1, 2, 3, … (HSF.IF.A.3) and asks kids to write them both ways, explicitly and recursively (HSF.BF.A.2). Arithmetic sequences are lines; geometric sequences are exponential growth.
The formula matters because listing breaks down fast. The 40th term of 3, 7, 11, … is 3 + 4 × 39 = 159, no list needed.
At the kitchen table
What to say, and what to skip
Say this
What do you do to get from one term to the next?
Add or multiply?Is it the same every time?
Check more than one pair.Why n − 1 and not n?
The first term hasn't had any jumps yet.Can you use your formula to get a term you already know?
That's the check.
Skip this
- Using n instead of n − 1.a₁ has zero jumps, so the 10th term has 9.
- Calling a sequence geometric because it grows fast.Divide neighbors. If the ratio isn't the same, it isn't geometric.
- Multiplying before raising to the power.2 · 3⁴ is 2 · 81, not 6⁴.
See it in action
Finding the 20th term of 5, 8, 11, 14, …
Find the common difference
8 − 5 = 3, 11 − 8 = 3, so d = 3Write the formula
aₙ = 5 + 3(n − 1)Plug in n = 20
5 + 3(19) = 62
Check with n = 4: 5 + 3(3) = 14 ✓.
Free practice sheet
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Parents also ask
What's a recursive formula?
A rule that says how to get the next term from the one before, like aₙ = aₙ₋₁ + 3 with a₁ = 5.
Can d or r be negative?
Yes. 20, 15, 10 has d = −5. 3, −6, 12 has r = −2, so the signs flip.
What if r is a fraction?
The terms shrink: 64, 32, 16 has r = 1/2.
Is every sequence arithmetic or geometric?
No. 1, 4, 9, 16 (the squares) is neither.