Geometric Sequence: worked example
| First term (a₁) | 2 |
|---|---|
| Common ratio (r) | 3 |
| Number of terms (n) | 8 |
| Last term (aₙ) | 4,374 |
| Sum of all terms | 6,560 |
Geometric Sequence: formula
Enter the first term, the common ratio, which is the number each term is multiplied by, and the number of terms. The last term is a1 times r to the power n minus 1, and the sum is a1 times one minus r to the n divided by one minus r, or n times a1 when r is 1. The ratio can be a fraction or negative.
A ratio between minus 1 and 1 produces shrinking terms.
Geometric Sequence: changing First term (a₁)
| First term (a₁) | Last term (aₙ) | Sum of all terms |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 2,187 | 3,280 |
| 2 (example) | 4,374 | 6,560 |
| 3 | 6,561 | 9,840 |
| 4 | 8,748 | 13,120 |
Geometric Sequence: compare input ranges
| Input | Tested from | To | Last term (aₙ): low input | Last term (aₙ): high input | Swing |
|---|---|---|---|---|---|
| Number of terms (n) | 7 | 9 | 1,458 | 13,122 | 11,664 (267%) |
| Common ratio (r) | 2.7 | 3.3 | 2,092.1 | 8,523.7 | 6,431.6 (147%) |
| First term (a₁) | 1 | 3 | 2,187 | 6,561 | 4,374 (100%) |
Geometric Sequence: individual input comparisons
| Input | Value | Last term (aₙ) | Sum of all terms |
|---|---|---|---|
| First term (a₁) | 0 | 0 | 0 |
| 1 | 2,187 | 3,280 | |
| 2 | 4,374 | 6,560 | |
| 3 | 6,561 | 9,840 | |
| 4 | 8,748 | 13,120 | |
| Common ratio (r) | 2.4 | 917.3 | 1,571.1 |
| 2.7 | 2,092.1 | 3,321.5 | |
| 3 | 4,374 | 6,560 | |
| 3.3 | 8,523.7 | 12,228.8 | |
| 3.6 | 15,672.8 | 21,700.1 | |
| Number of terms (n) | 6 | 486 | 728 |
| 7 | 1,458 | 2,186 | |
| 8 | 4,374 | 6,560 | |
| 9 | 13,122 | 19,682 | |
| 10 | 39,366 | 59,048 |
Geometric Sequence: combined input changes
| 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 |
Geometric Sequence: which assumptions move the result?
Geometric Sequence: values and definitions
| Quantity and definition | Value used |
|---|---|
| First term (a₁) — The starting value of the sequence. | 2 |
| Common ratio (r) — The fixed multiplier applied at every step. A ratio between −1 and 1 shrinks toward zero. | 3 |
| Number of terms (n) — How many terms to include, from 1 to 1,000. | 8 |
| Last term (aₙ) | 4,374 |
| Sum of all terms | 6,560 |
Geometric Sequence: half and double Number of terms (n)
| Number of terms (n) | Last term (aₙ) |
|---|---|
| 4 | 54 |
| 8 | 4,374 |
| 16 | 28,697,814 |
Common error: 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.
Terms grow very quickly for ratios above 1, so results can overflow. If the ratio is exactly 1 the sequence is constant, and if it is negative the terms alternate in sign. The finite sum formula divides by one minus r, which is why r equal to 1 is handled separately.
Real growth is rarely perfectly geometric.
A negative ratio alternates signs
For the assumed terms 3, −6, 12 and −24, the common ratio is −2 and the four-term sum is −15. A negative ratio changes the signs; it does not mean a constant subtraction. The finite geometric-sum formula works with this ratio even though an infinite series with absolute ratio greater than one does not converge.
At ratio one, add n copies of the first term rather than dividing by 1 − r. Distinguish a request for the nth term from a request for the sum through that term before using the answer. For the same first term three, choosing ratio one gives 3, 3, 3, 3 and a four-term sum of twelve.
Choosing ratio zero gives 3, 0, 0, 0 and sum three. The closed fraction with denominator one minus the ratio needs its special case at one, even though the finite sum itself is perfectly defined. A term number must be a whole counting position.
A fractional exponent of a negative ratio is a different problem from listing a real sequence at integer positions.
Algebra & Arithmetic calculators: Percent Difference Calculator, Proportion Calculator, Roman Numeral Calculator, Completing the Square Calculator, Dot Product Calculator, Mixed Number to Improper Fraction Calculator.
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.
Look at the ratio first. Above 1 in size, the last term drives the sum. Below 1, the first few terms do.
Compare with the arithmetic sequence calculator, and apply the idea on the exponential growth and compound interest pages. The exponent and logarithm calculators solve for the number of terms, and the Fibonacci page shows a growth pattern that is nearly geometric. The rule of 72 estimates doubling time.
What is a geometric sequence?
A sequence in which each term is the previous term multiplied by a fixed ratio, such as 2, 6, 18, 54 with a ratio of 3.
Can a geometric series have an infinite sum?
Yes, when the ratio is between minus 1 and 1, the sum of infinitely many terms equals the first term divided by one minus the ratio.
How is it related to compound interest?
Compound interest multiplies the balance by one plus the rate each period, so the balances form a geometric sequence.