Example Data Table
| First Term |
Ratio |
Start |
Terms |
Mode |
Expected Sum |
| 3 |
2 |
1 |
6 |
Finite |
189 |
| 10 |
0.5 |
1 |
8 |
Finite |
19.921875 |
| 5 |
0.25 |
1 |
Not required |
Infinite |
6.666667 |
| 4 |
-0.5 |
2 |
5 |
Finite |
-1.375 |
Formula Used
Selected first term: A = a × rstart - 1
Finite sum when r is not 1: S = A × (1 - rn) ÷ (1 - r)
Finite sum when r is 1: S = A × n
Infinite sum when |r| is less than 1: S∞ = A ÷ (1 - r)
Last selected term: L = A × rn - 1
Average selected term: Average = S ÷ n
How to Use This Calculator
Choose finite sum when you know how many terms to add. Choose infinite sum when the ratio has an absolute value below one.
Enter the first term and common ratio. Add the start position if the sum begins later in the sequence.
Set the number of terms for finite work. Select decimal places for cleaner output. Add a target sum when you want comparison feedback.
Press Calculate to view the answer above the form. Use CSV or PDF buttons to save the report.
Geometric Sequence Sum Guide
What This Calculator Does
A geometric sequence multiplies each term by the same ratio. This calculator studies that pattern from many angles. It finds finite sums, infinite sums, last terms, selected starting terms, and average term value. It also shows a compact step trail, so the answer is easier to audit.
Why The Sum Matters
Geometric sums appear in savings plans, loan models, population estimates, signal decay, game rewards, and repeated discounts. A small ratio change can create a large change in the total. That is why the tool includes rounding, target comparison, and partial sequence rows. You can test growth and decay without rewriting formulas each time.
Finite Sum Logic
A finite sum adds a fixed number of terms. The first selected term may be the original first term, or it may start later. When the ratio is not one, the calculator uses the closed form. It avoids long repeated addition. When the ratio equals one, every selected term is equal, so the sum becomes one term multiplied by the count.
Infinite Sum Logic
An infinite geometric sum works only when the absolute value of the ratio is less than one. In that case, later terms shrink toward zero. The total approaches a stable limit. When the ratio is one, negative one, or larger in size, the infinite sum does not settle. The calculator marks that condition clearly.
Interpreting Results
The nth term shows the last selected term in the finite block. The average term divides the sum by the number of selected terms. The target comparison helps check whether a planned total has been reached. The generated rows show early terms and running totals. These rows are useful for lessons and quick validation.
Good Input Practice
Use decimals carefully. Enter negative ratios when signs alternate. Use a positive whole number for terms. Keep rounding high when results are sensitive. Compare the displayed formula steps with the example table before using outputs in reports. For finance or engineering work, verify assumptions with the method required by your course, office, or standard.
Reusable Output Tips
Export files after checking inputs. Name downloads clearly. Store assumptions beside results, so future readers understand the selected range and ratio.
FAQs
1. What is a geometric sequence?
A geometric sequence is a list where each term is found by multiplying the previous term by the same ratio. The ratio can be positive, negative, fractional, or decimal.
2. What is a finite geometric sum?
A finite geometric sum adds a limited number of terms. You enter the first term, ratio, start position, and term count. The calculator then applies the closed formula.
3. When does an infinite geometric sum exist?
An infinite geometric sum exists when the absolute value of the common ratio is less than one. Then later terms shrink, and the total approaches a fixed limit.
4. What happens when the ratio equals one?
When the ratio equals one, every selected term is the same. The finite sum is the selected first term multiplied by the number of terms.
5. Can I start the sum from a later term?
Yes. Enter a start position greater than one. The tool first finds that selected term, then calculates the sum from that position forward.
6. Why is my infinite sum marked as not convergent?
The infinite sum is not convergent when the absolute value of the ratio is one or greater. The terms do not shrink toward zero fast enough.
7. What does the target sum field do?
The target sum field compares your calculated total with a desired value. It reports whether the result is equal, above, or below the target.
8. What do the download buttons include?
The CSV and PDF downloads include inputs, main outputs, formula steps, behavior notes, target comparison, and generated term rows for review.