Arithmetic and geometric sequences become much easier when you use a consistent decision process. This guide shows how to identify the pattern, write explicit and recursive formulas, find missing terms, check your work, and translate common sequence word problems into mathematics. Use the tracking checkpoints and practice sets to revisit the skill as your algebra or precalculus course becomes more advanced.
Overview
A sequence is an ordered list of numbers. Each number is called a term, and the position of a term is represented by n. For example, in the sequence 4, 7, 10, 13, the first term is 4, so a1 = 4, and the fourth term is 13, so a4 = 13.
The two patterns covered here are:
- Arithmetic sequence: the same number is added or subtracted each time.
- Geometric sequence: each term is multiplied by the same nonzero number.
Before choosing a formula, compare consecutive terms. Subtract neighboring terms to test for a common difference. If subtraction does not produce a constant result, divide a term by the previous term to test for a common ratio. This simple check prevents a common error: using an arithmetic formula for a geometric pattern, or vice versa.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Pattern | Add or subtract a constant | Multiply by a constant |
| Key value | Common difference, d | Common ratio, r |
| Explicit formula | an = a1 + (n − 1)d | an = a1(r)n−1 |
| Recursive formula | an = an−1 + d | an = an−1 × r |
The explicit formula finds any term directly. The recursive formula finds a term from the preceding term, so it is useful for showing how the sequence is built.
What to track
1. Track the pattern before the formula
For an arithmetic sequence such as 12, 8, 4, 0, subtract consecutive terms:
8 − 12 = −4, 4 − 8 = −4, and 0 − 4 = −4.
The common difference is d = −4. For a geometric sequence such as 5, 15, 45, 135, divide consecutive terms:
15 ÷ 5 = 3, 45 ÷ 15 = 3, and 135 ÷ 45 = 3.
The common ratio is r = 3. A negative difference or ratio is allowed. For example, 20, −10, 5, −2.5 is geometric with ratio −1/2.
2. Track the first term, position, and pattern value
Write down a1, the requested position n, and either d or r before substituting. This is especially helpful when a problem begins at a position other than 1. If a sequence gives a4 = 17 and d = 3, you cannot automatically use 17 as a1. Instead, work backward: a1 = 17 − 3(3) = 8.
3. Track formulas in both forms
Arithmetic example: Find an explicit formula for 6, 10, 14, 18, ...
- Identify a1 = 6 and d = 4.
- Substitute into an = a1 + (n − 1)d.
- Write an = 6 + 4(n − 1), or simplify to an = 4n + 2.
The recursive form is a1 = 6 and an = an−1 + 4.
Geometric example: Find the eighth term of 3, 6, 12, 24, ...
- Identify a1 = 3 and r = 2.
- Use an = a1(r)n−1.
- Calculate a8 = 3(2)7 = 384.
The recursive form is a1 = 3 and an = 2an−1.
4. Track missing terms and word-problem quantities
For a missing term, first decide whether the blanks follow equal additions or equal multiplications. In 7, __, 19, the difference is 6, so the missing term is 13. In 4, __, 36, the positive ratio is 3, so the missing term is 12.
For word problems, define the first term and the time interval. A theater has 120 seats in the first row and adds 8 seats in each successive row. The number of seats per row is arithmetic: an = 120 + 8(n − 1). If an online video has 500 views in its first hour and triples each hour, the model is geometric: vn = 500(3)n−1. Always state what n counts—rows, hours, payments, or another interval.
5. Use a spreadsheet as a check, not a substitute for reasoning
In a spreadsheet, place term numbers 1, 2, 3, and so on in one column. For an arithmetic sequence, enter the first term in the next column and use a formula such as =previous_cell+$B$1, where cell B1 stores the common difference. For a geometric sequence, use =previous_cell*$B$1, where B1 stores the ratio. Compare the generated terms with your hand-written formula. If the values disagree, check the exponent n − 1 and confirm that the starting term is aligned with position 1.
Cadence and checkpoints
Sequence skills improve when you revisit the same decision process rather than memorize isolated formulas. Use these checkpoints:
- After learning: classify five short sequences by testing differences and ratios.
- During weekly practice: write one explicit formula and one recursive formula, then verify two generated terms.
- Before a quiz or test: solve a mixed set containing missing terms, term finding, and word problems without being told the type of sequence.
- Monthly or at the end of a unit: review your error log and repeat problems involving the mistake you make most often.
Keep a small sequence tracker with these columns: problem, sequence type, a1, d or r, formula, requested term, answer, and error note. This makes progress visible and helps separate pattern-identification errors from arithmetic errors.
How to interpret changes
An arithmetic sequence changes by a fixed amount, so its terms have a constant first difference. Its graph, when term number is plotted against term value, follows a linear pattern. A geometric sequence changes by a fixed factor, so its terms have a constant ratio. Its growth may be rapid when the absolute value of the ratio is greater than 1, or it may move toward zero when the absolute value is between 0 and 1.
Do not confuse “increasing” with “geometric” or “decreasing” with “arithmetic.” The sequence 2, 4, 6, 8 increases arithmetically, while 2, 4, 8, 16 increases geometrically. Also watch for alternating signs. A negative geometric ratio produces terms that switch positive and negative, but the ratio remains constant.
Graduated practice
- Find the common difference and the tenth term of 9, 13, 17, 21, ...
- Find the common ratio and the sixth term of 2, 6, 18, 54, ...
- Write an explicit formula for 25, 20, 15, 10, ...
- A savings plan starts at $40 in week 1 and increases by $5 each week. How much is saved in week 12?
- A population starts at 800 and decreases by 10% each year. Write a formula for the population after n years, assuming the starting value is at year 1.
Answers and explanations: 1. d = 4, so a10 = 9 + 9(4) = 45. 2. r = 3, so a6 = 2(3)5 = 486. 3. a1 = 25 and d = −5, so an = 25 − 5(n − 1). 4. The plan is arithmetic: 40 + 5(11) = $95. 5. A 10% decrease leaves 90%, or 0.9, each year, so an = 800(0.9)n−1 if week-style indexing begins at year 1.
When to revisit
Return to this guide whenever a new problem introduces a different starting position, a negative ratio, a missing term, or a real-world quantity that changes over time. It is also worth revisiting before a precalculus assessment, because sequence notation often connects to function notation, exponential models, and series.
Use this final checklist before submitting a solution:
- Did I test subtraction and division to classify the sequence?
- Did I identify the correct first term and position?
- Did I use n − 1 in the explicit formula?
- Did I describe what the variable represents in a word problem?
- Did I generate at least one additional term to check the formula?
If the difficulty is actually with rearranging equations, function notation, or an exponential model, continue with the related guides on function notation, exponential equations, and math word problems. Record the next error you find in your tracker, then repeat a short mixed practice set later in the week.
