Calculator Input
Choose a rule type, enter the values, and calculate the first six terms. Results appear above this form after submission.
Example Data Table
These sample patterns show how sequence rules can support study plans, promotion milestones, job applications, and career growth targets.
| Scenario | Rule Type | Rule | First 6 Terms |
|---|---|---|---|
| Weekly job applications | Arithmetic | a₁ = 5, d = 3 | 5, 8, 11, 14, 17, 20 |
| Networking outreach growth | Geometric | a₁ = 2, r = 2 | 2, 4, 8, 16, 32, 64 |
| Skill points by project cycle | Explicit | aₙ = n² + 1 | 2, 5, 10, 17, 26, 37 |
| Progress review checkpoints | Recursive | a₁ = 4, p = 1.5, q = 2 | 4, 8, 14, 23, 36.5, 56.75 |
Formula Used
1) Explicit Formula
Enter a direct nth-term rule. The calculator substitutes n = 1 through 6 into your formula and returns the first six values.
2) Arithmetic Sequence
aₙ = a₁ + (n - 1)d
Here, a₁ is the first term and d is the constant difference.
3) Geometric Sequence
aₙ = a₁r⁽ⁿ⁻¹⁾
Here, a₁ is the first term and r is the constant ratio.
4) Quadratic Rule
aₙ = an² + bn + c
This helps when growth follows a curved pattern rather than a constant step.
5) Recursive Rule
aₙ = p·aₙ₋₁ + q
Start from a₁. Each new term comes from multiplying the previous term by p and then adding q.
How to Use This Calculator
- Select the sequence mode that matches your rule.
- Enter either a direct formula or the required sequence values.
- Click Find the First 6 Terms.
- Review the six terms, the trend, the sum, and the average.
- Check the differences to understand the pattern more deeply.
- Use the Plotly graph to spot growth, decline, or instability.
- Download CSV for spreadsheets or PDF for clean reports.
- Apply the output to quotas, milestones, learning plans, or forecasting tasks.
FAQs
1) What does this calculator do?
It finds the first six terms of a sequence from a chosen rule. You can use explicit formulas, arithmetic rules, geometric rules, quadratic rules, or a recursive pattern.
2) Which sequence type should I choose?
Choose arithmetic for fixed additions, geometric for fixed multiplications, quadratic for curved growth, recursive for step-by-step rules, and explicit when you already know the nth-term expression.
3) Can I type powers in the explicit formula?
Yes. Use the caret symbol for powers, such as n^2 or 3*n^3. Use explicit multiplication symbols too, like 2*n instead of 2n.
4) How can this help with career planning?
You can model application targets, study hours, networking goals, sales quotas, savings plans, or promotion milestones. Sequences turn abstract growth plans into measurable steps.
5) Why does the calculator show first and second differences?
Differences reveal pattern behavior. Constant first differences often suggest arithmetic growth. Constant second differences often suggest quadratic behavior. They help you verify whether your chosen rule makes sense.
6) Why only the first six terms?
Six terms are enough to reveal most common sequence patterns clearly. They also fit neatly into tables, reports, and planning sheets without overwhelming the reader.
7) Can I export the results?
Yes. Use the CSV button for spreadsheet analysis and the PDF button for printable summaries. Both export options capture your current six-term result set.
8) What mistakes should I avoid?
Avoid missing multiplication symbols, unsupported characters, and wrong rule selection. Double-check whether your pattern changes by addition, multiplication, or a formula-based expression before calculating.