Calculator Inputs
Example Data Table
| First Term | Difference | Index n | Nth Term | Sum First n Terms |
|---|---|---|---|---|
| 5 | 3 | 10 | 32 | 185 |
| 12 | -2 | 8 | -2 | 40 |
| 1.5 | 0.5 | 6 | 4 | 16.5 |
Formula Used
Nth term: aₙ = a₁ + (n - 1)d
Sum of first n terms: Sₙ = n / 2 × [2a₁ + (n - 1)d]
Common difference from two terms: d = (aᵩ - aₚ) / (q - p)
First term from a known term: a₁ = aₚ - (p - 1)d
Here, a₁ is the first term. d is the common difference.
n is the required term position.
How to Use This Calculator
- Enter the first term and common difference.
- Enter the term index you want to calculate.
- Add a range when you need a selected sum.
- Use two known indexed terms when first term is unknown.
- Press calculate to view results above the form.
- Download the table as CSV or export a PDF report.
Arithmetic Sequences in Maths
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers with a fixed gap. This gap is called the common difference. Each new term is made by adding the same value to the previous term. The sequence may rise, fall, or stay constant. A positive difference makes values increase. A negative difference makes values decrease.
Why This Tool Is Useful
This calculator helps you study many sequence questions in one place. You can find the nth term, first term, common difference, range sum, and running totals. You can also build a table for many terms. The graph shows the pattern clearly. Straight movement on the graph confirms the constant difference.
Advanced Term Solving
Sometimes the first term is not given. You may only know two terms from different positions. In that case, enter both indexes and values. The calculator finds the common difference. Then it works backward to find the first term. This is helpful in exams, finance schedules, number patterns, and lesson planning.
Understanding Sums
A sequence sum adds terms together. The first n terms can be added with a direct formula. A selected range can also be added. This is useful when you only need part of the sequence. Running sums help you see how totals grow term by term.
Checking Your Work
Use the example table before entering your own values. Compare the nth term with the generated row. Check the graph for a steady line. Export the CSV file for spreadsheets. Use the PDF report for homework, teaching notes, or quick records.
FAQs
1. What is an arithmetic sequence?
An arithmetic sequence is a number pattern where each term changes by the same fixed amount. That fixed amount is called the common difference.
2. What is the common difference?
The common difference is the value added to one term to get the next term. It can be positive, negative, zero, or decimal.
3. How do I find the nth term?
Use the formula aₙ = a₁ + (n - 1)d. Enter the first term, common difference, and the required index.
4. Can the sequence decrease?
Yes. A sequence decreases when the common difference is negative. For example, 20, 15, 10, and 5 has a difference of -5.
5. Can I find a sequence from two known terms?
Yes. Enter both known term indexes and their values. The calculator can infer the common difference and first term from those inputs.
6. What does the sum formula calculate?
The sum formula calculates the total of the first n terms. It saves time because you do not need to add each term manually.
7. Why is the graph a straight pattern?
Arithmetic sequences change by a constant amount. Because the change is steady, the plotted points follow a straight-line pattern.
8. What can I export from this calculator?
You can export generated terms and running sums as a CSV file. You can also download a PDF report of main results.