Calculator
Example Data Table
| Known term n | Value at n | Known term m | Value at m | Common ratio | First term |
|---|---|---|---|---|---|
| 2 | 6 | 5 | 48 | 2 | 3 |
| 1 | 81 | 5 | 1 | 1/3 or -1/3 | 81 |
| 3 | 20 | 6 | 160 | 2 | 5 |
| 2 | -8 | 5 | 64 | -2 | 4 |
Formula Used
The geometric sequence term formula is:
Tn = a × rn - 1
Here, a is the first term, r is the common ratio, and n is the term position.
When two terms are known, the common ratio is found with:
r = (Tm / Tn)1 / (m - n)
After the ratio is known, the first term is:
a = Tn / rn - 1
The finite sum of the first N terms is:
SN = a(1 - rN) / (1 - r), when r is not 1.
When r equals 1, the finite sum is SN = a × N.
The infinite sum is a / (1 - r), but only when |r| < 1.
How To Use This Calculator
- Select a calculation mode.
- Use two known terms when you need to solve for the first term and ratio.
- Use direct analysis when the first term and ratio are already known.
- Enter the term position that you want to inspect.
- Choose how many generated terms should appear in the table.
- Set decimal precision for rounded output.
- Press Calculate.
- Review the result section above the form.
- Use CSV or PDF download for saved results.
Understanding First Term And Common Ratio
What This Calculator Does
A geometric sequence changes by a fixed multiplier. That multiplier is the common ratio. The first term starts the sequence. When two terms are known, both values can often be recovered. This calculator solves that task and then builds a working sequence table.
The tool accepts two known terms, their positions, and a display length. It finds every real ratio that can match the data. It also handles the special case where an even position gap allows a positive and a negative ratio. After a valid ratio is found, the first term is calculated from the chosen term position.
Why First Term And Ratio Matter
The first term and common ratio describe the whole pattern. They let you predict any later term. They also help check growth, decay, alternating signs, and convergence. This is useful in algebra, finance, science, and computer modeling. Many problems hide the first term. They only give two later values. In those cases, this calculator shows the missing structure.
Sequence Analysis Features
The result includes the nth term rule, finite sum, infinite sum status, ratio classification, and a generated table. The finite sum is useful when you need the total of several terms. The infinite sum is shown only when the absolute value of the ratio is below one. That condition means the sequence approaches a stable total.
The calculator also supports direct analysis. Enter a first term and a ratio when those values are already known. Then choose the term position you want to inspect. The output explains the selected term, total, and behavior.
Practical Use Cases
Students can verify homework steps before submitting answers. Teachers can create examples for lessons. Analysts can model repeated percentage change. Engineers can study scaling patterns. Business users can estimate compounding growth or decline.
Use the download buttons after calculation. The CSV file is helpful for spreadsheets. The report file is useful for records and sharing. Always review negative ratio cases carefully. A negative ratio alternates signs. That may be correct for algebra, but it may not fit every real world model.
For best accuracy, enter positions as positive integers. Keep term values exact when possible, especially when roots or fractional ratios are expected too.
FAQs
1. What is a first term?
The first term is the starting value of a geometric sequence. It is usually written as a. Once it is known with the common ratio, every later term can be calculated.
2. What is a common ratio?
The common ratio is the fixed multiplier between consecutive terms. Divide any term by the previous term to find it, when both terms are non-zero.
3. Can the common ratio be negative?
Yes. A negative common ratio creates alternating signs. For example, 4, -8, 16, -32 has a common ratio of -2.
4. Why can two answers appear?
Two answers can appear when the gap between known term positions is even. A positive and negative ratio may both create the same known terms.
5. What happens when the ratio is one?
Every term stays equal to the first term. The sequence is constant. The finite sum is simply the first term multiplied by the number of terms.
6. When does the infinite sum exist?
The infinite sum exists only when the absolute value of the common ratio is less than one. Then the terms shrink toward zero.
7. Can I use decimals?
Yes. The calculator accepts decimal term values, first terms, and ratios. Increase decimal precision when working with small values or roots.
8. What is the generated table for?
The generated table shows term values and running sums. It helps verify the sequence pattern and makes exported reports easier to review.