Answer first
What this calculator tells you
Find the last term and the sum of a geometric sequence from its first term, common ratio and length. Total a series that grows or shrinks by a fixed multiple, such as doubling or a bouncing ball. Formula: nth term = a₁ × rⁿ⁻¹; sum of n terms = a₁(1 − rⁿ) ÷ (1 − r), or n × a₁ when r = 1. At the worked-example inputs, the last term (aₙ) is 4,374. Holding every other input steady, moving first term (a₁) from 0 to 4 moves the result from 0 to 8,748.
Transparent method
The formula
Total a series that grows or shrinks by a fixed multiple, such as doubling or a bouncing ball.
Worked example
Example inputs
How to interpret the result
A geometric sequence multiplies by the same ratio each time, so it starts slowly and then dominates the total. Eight terms from 2 with a ratio of 3 end at 4,374, and the last term alone is two thirds of the 6,560 sum. That pattern is why compounding, doubling and viral spread surprise people, and why a ratio between negative one and one does the opposite and settles toward zero.
At the worked-example inputs the last term (aₙ) is 4,374. It rises with number of terms (n), common ratio (r) and first term (a₁).
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects.
Before you rely on it
What to check
Look at the ratio first. Above 1 in size, the last term drives the sum. Below 1, the first few terms do.
The common error
Where people go wrong with geometric sequence calculator
Treating a percentage growth rate as the ratio. Growth of 5 percent per step means a ratio of 1.05, not 5, and entering 5 overstates the last term enormously.
Sensitivity evidence
How first term (a₁) changes the last term (aₙ)
Holding every other input at the worked-example value, moving first term (a₁) from 0 to 4 moves the last term (aₙ) from 0 to 8,748: a spread of 8,748, or 200% of the worked-example result.
| First term (a₁) | Last term (aₙ) | Sum of all terms |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 2,187 | 3,280 |
| 2worked example | 4,374 | 6,560 |
| 3 | 6,561 | 9,840 |
| 4 | 8,748 | 13,120 |
Every input, tested
Which input moves the last term (aₙ) most
Of the 3 inputs, number of terms (n) moves the last term (aₙ) most (11,664 across the range tested) and first term (a₁) moves it least (4,374).
| Input | Tested from | To | Last term (aₙ) at each end | Swing |
|---|---|---|---|---|
| Number of terms (n) | 7 | 9 | 1,458 to 13,122 | 11,664 (267%) |
| Common ratio (r) | 2.7 | 3.3 | 2,092.1 to 8,523.7 | 6,431.6 (147%) |
| First term (a₁) | 1 | 3 | 2,187 to 6,561 | 4,374 (100%) |
Two variables at once
Last term (aₙ) by first term (a₁) and common ratio (r)
Across the grid the last term (aₙ) runs from 0 to 31,345.7. Moving first term (a₁) from 0 to 4 shifts it by 8,748 at the middle column, and moving common ratio (r) from 2.4 to 3.6 shifts it by 14,755.5 at the middle row, so common ratio (r) is the bigger lever here.
| First term (a₁) \ Common ratio (r) | 2.4 | 3 | 3.6 |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 458.6 | 2,187 | 7,836.4 |
| 2 | 917.3 | 4,374 | 15,672.8 |
| 3 | 1,375.9 | 6,561 | 23,509.2 |
| 4 | 1,834.6 | 8,748 | 31,345.7 |
The highlighted cell is the worked example: 4,374.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| First term (a₁) | 2 | The starting value of the sequence. |
| Common ratio (r) | 3 | The fixed multiplier applied at every step. A ratio between −1 and 1 shrinks toward zero. |
| Number of terms (n) | 8 | How many terms to include, from 1 to 1,000. |
| Last term (aₙ) | 4,374 | |
| Sum of all terms | 6,560 | |
Inputs, definitions and assumptions
First term (a₁)
The starting value of the sequence. The prefilled worked-example value is 2.
Common ratio (r)
The fixed multiplier applied at every step. A ratio between −1 and 1 shrinks toward zero. The prefilled worked-example value is 3.
Number of terms (n)
How many terms to include, from 1 to 1,000. The prefilled worked-example value is 8.
How to use this calculator
- 1Verify the inputs. Gather first term (a₁), common ratio (r) and number of terms (n) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the last term (aₙ) at 4,374. Store your own version of it as Scenario A.
- 3Test one change. Start with number of terms (n), the input with the biggest effect here: moving number of terms (n) from 7 to 9 takes the last term (aₙ) from 1,458 to 13,122, a swing of 267% of the worked-example figure.
- 4Check the extremes. At half the example number of terms (n) (4) the last term (aₙ) is 54; at double (16) it is 28,697,814.
People also ask
Frequently asked questions
How do you calculate geometric sequence?
nth term = a₁ × rⁿ⁻¹; sum of n terms = a₁(1 − rⁿ) ÷ (1 − r), or n × a₁ when r = 1. At the worked-example inputs the last term (aₙ) is 4,374.
What does the geometric sequence result mean?
Total a series that grows or shrinks by a fixed multiple, such as doubling or a bouncing ball. At the worked-example inputs the last term (aₙ) is 4,374. It rises with number of terms (n), common ratio (r) and first term (a₁).
How much does first term (a₁) change the last term (aₙ)?
Holding every other input at the worked-example value, moving first term (a₁) from 0 to 4 moves the last term (aₙ) from 0 to 8,748, a spread of 8,748.
What are the limits of this geometric sequence calculator?
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects. The tables on this page test first term (a₁) only from 0 to 4; a value outside that range is not tabulated here.
Which input moves the last term (aₙ) most in the geometric sequence calculator?
Ranked by how far each moves the last term (aₙ) across the range tested: number of terms (n) (11,664, 267%), common ratio (r) (6,431.6, 147%) and first term (a₁) (4,374, 100%).
If I double number of terms (n) in the geometric sequence calculator, does the last term (aₙ) double?
Doubling it from 8 to 16 takes the last term (aₙ) from 4,374 to 28,697,814, which is 6561.00 times the worked-example figure. So the result grows faster than the input does. Halving it to 4 gives 54.
How much does common ratio (r) matter in the geometric sequence calculator?
The worked example uses 3. With the other inputs left at the worked example, moving common ratio (r) from 2.7 to 3.3 takes the last term (aₙ) from 2,092.1 to 8,523.7, a swing of 147% of the worked-example figure.
How much does number of terms (n) matter in the geometric sequence calculator?
The worked example uses 8. With the other inputs left at the worked example, moving number of terms (n) from 7 to 9 takes the last term (aₙ) from 1,458 to 13,122, a swing of 267% of the worked-example figure.
Which inputs change the sum of all terms in the geometric sequence calculator?
At the worked-example inputs it is 6,560. First term (a₁) takes it from 3,280 to 9,840, common ratio (r) takes it from 3,321.5 to 12,228.8 and number of terms (n) takes it from 2,186 to 19,682.
When should I use a weighted average instead of a plain average?
When the values don't all count equally: exam scores worth different percentages of a grade, or portfolio returns weighted by how much is invested in each asset. A plain average silently treats every value as equally important, which is often not true.
Sources and evidence
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