Calculated sequence
Term table
| Position | Index | Exact | Approximate |
|---|
Sequence graph
Step-by-step calculations
Sequence settings
Formula used
Arithmetic sequences add one constant difference between consecutive terms. Geometric sequences multiply each term by one constant ratio. Explicit rules calculate every term directly from its index.
Recursive rules require one or two initial values. First-order rules use only the previous term. Second-order rules use the previous two calculated terms directly.
How to use this calculator
Select the sequence type matching your given rule. Enter the required values and starting index. Keep six terms selected for the requested answer format.
Choose an exact or decimal primary display. Press calculate to generate terms and supporting analysis. Review every substitution before copying or exporting results carefully.
Worked examples
| Sequence rule | Inputs | First six terms |
|---|---|---|
| Arithmetic | a₁ = 2, d = 3 | 2, 5, 8, 11, 14, 17 |
| Geometric | a₁ = 2, r = 3 | 2, 6, 18, 54, 162, 486 |
| Explicit | a(n) = n² | 1, 4, 9, 16, 25, 36 |
| Recursive | a₁ = 1, a₂ = 1, next = prev1 + prev2 | 1, 1, 2, 3, 5, 8 |
Common mistakes
Check whether indexing begins at zero or one. Apply exponents before multiplication and addition. Enter parentheses whenever the intended grouping could become unclear.
Do not confuse common differences with common ratios. Recursive formulas need valid initial values before calculation. Review undefined divisions, roots, or logarithms before accepting answers.
Frequently asked questions
What are the first six terms?
They are the earliest six values generated by a rule. Their indexes commonly run from one through six. Some problems instead begin indexing from zero by convention.
How does an arithmetic sequence work?
An arithmetic sequence adds the same difference repeatedly. Subtract consecutive terms to verify that difference. Equal differences confirm an arithmetic pattern across all terms.
How does a geometric sequence work?
A geometric sequence multiplies by one constant ratio. Divide consecutive nonzero terms to test that ratio. Equal ratios confirm a geometric pattern across all terms.
Can the sequence start at n = 0?
Yes, change the starting index field to zero. Explicit formulas then evaluate zero as the first index. Arithmetic and geometric starts remain tied to entered values.
Can I enter fractions?
Explicit formulas accept fractions using normal division notation. Numeric input fields also accept decimal equivalents. Exact formatting displays simple rational values whenever practical here.
What does prev1 mean?
Prev1 represents the immediately preceding calculated sequence term. It supports first-order and second-order recursive rules. Prev2 represents the term appearing two positions earlier there.
Why is a term undefined?
A formula may divide by zero at that index. Roots or logarithms may also leave real numbers. Adjust the rule, domain, or starting index accordingly first.
How is the sequence classified?
The calculator compares differences, ratios, signs, and direction. Floating calculations use a small numerical tolerance. Classification summarizes patterns without replacing formal mathematical proof requirements.
Can I calculate more than six terms?
Yes, the term count supports up to fifty. Six remains the default for this specific task. Larger counts provide broader graphs and longer downloadable tables.