Geometric Sequence Calculator
Find the nth term, the sum of the first n terms, the sum to infinity or a term's position in a geometric sequence. Shows the formula and each step.
Geometric Sequence Calculator
Calculate terms, sum, and other properties of geometric sequences. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Sequence Parameters
Calculation Type
Correcting A Common Mistake: The Common Ratio Can Be 1 Or 0
Many students think a geometric sequence requires a common ratio that is not 0 or 1. That is false. A sequence with r = 1 is a constant sequence: every term equals the first term. The geometric sequence calculator handles that case correctly. A sequence with r = 0 produces the first term followed by all zeros, which is also geometric. The real requirement is that the ratio between consecutive terms must be constant. Zero and one are constant values, so they qualify.
- Definition: A sequence where each term after the first is the previous term multiplied by a fixed, non-zero number called the common ratio (r).
- General Term: aₙ = a₁ × r^(n-1), where n is a positive integer.
- Sum of First n Terms: Sₙ = a₁ × (1 - rⁿ) / (1 - r) when r ≠ 1; Sₙ = n × a₁ when r = 1.
- Sum to Infinity: S∞ = a₁ / (1 - r), valid only when |r| < 1.
- Source: OpenStax Algebra and Trigonometry 2e, sections 13.3 and 13.4.
How To Use Each Calculation Mode
Enter the first term (a₁) and the common ratio (r). Then choose what to calculate from the dropdown menu. The calculator shows the result and, when you select the option, the step-by-step working.
Find The Nth Term
Provide the term position n. The calculator applies aₙ = a₁ × r^(n-1) and returns the value of that term. Use this mode when you know the position and need the term itself.
Sum Of First N Terms
Enter the number of terms n. The calculator uses Sₙ = a₁ × (1 - rⁿ) / (1 - r), or Sₙ = n × a₁ if r = 1. This gives the total of the first n terms.
Sum To Infinity
The calculator checks whether |r| < 1. If the condition is met, it applies S∞ = a₁ / (1 - r). If |r| ≥ 1, it returns an error because the series does not converge to a finite sum.
Find Position Of A Term
Enter a term value. The calculator solves aₙ = a₁ × r^(n-1) for n using logarithms. It reports whether the value appears at an integer position, a full term, or falls between two terms. Negative ratios cause alternating signs, which the calculator handles by checking sign matches.
Generate A Sequence
Specify how many terms to generate (1 to 50). The calculator lists each term with its calculation, shows the running sum, and displays a chart of the sequence.
The Formulas The Calculator Uses
All formulas come directly from the definition of a geometric sequence. The nth term relation works for any real r and any positive integer n. The partial sum relation fails at r = 1 because the denominator becomes zero, so the calculator substitutes the special-case relation Sₙ = n × a₁.
The sum to infinity relation depends on convergence. An infinite geometric series converges only when |r| < 1. As n increases, the term rⁿ approaches zero, and the partial sum approaches a₁/(1 - r). When |r| ≥ 1, the terms do not approach zero, so the series diverges and has no finite sum.
These are the same relations found in OpenStax Algebra and Trigonometry 2e, sections 13.3 and 13.4, and in Stewart Calculus, section 11.2.
Worked Example: Nth Term And Sum
Take the sequence 3, 6, 12, 24, 48, ... The first term a₁ = 3 and the common ratio r = 2.
Finding The 10th Term
Use a₁₀ = 3 × 2^(10-1) = 3 × 2⁹ = 3 × 512 = 1,536. The 10th term is 1,536.
Sum Of The First 10 Terms
Use S₁₀ = 3 × (1 - 2¹⁰) / (1 - 2) = 3 × (1 - 1,024) / (-1) = 3 × (-1,023) / (-1) = 3 × 1,023 = 3,069. The sum of the first 10 terms is 3,069.
For a decaying sequence like 100, 50, 25, 12.5, 6.25, ... where a₁ = 100 and r = 0.5, the 10th term is 100 × 0.5⁹ = 100 × 0.001953125 = 0.1953125. The sum of the first 10 terms is 100 × (1 - 0.5¹⁰) / (1 - 0.5) = 100 × (1 - 0.0009765625) / 0.5 = 100 × 0.9990234375 / 0.5 = 199.8046875.
When The Sum To Infinity Exists
The sum to infinity exists only when |r| < 1. This is the convergence condition for an infinite geometric series. Under that condition, the terms get smaller and smaller, approaching zero, and the partial sums approach a finite limit. If |r| ≥ 1, the series diverges, the terms do not shrink, and the sum either grows without bound (r > 1) or continues to oscillate (r ≤ -1).
For example, the series 10 + 5 + 2.5 + 1.25 + ... has r = 0.5, so it converges to S∞ = 10 / (1 - 0.5) = 20. The series 2 + 4 + 8 + 16 + ... has r = 2, so it diverges, the sum is infinite. The series 3 - 6 + 12 - 24 + ... has r = -2, and |r| = 2 ≥ 1, so it also diverges.
This distinction matters for applications like calculating the total distance traveled by a bouncing ball or the present value of a perpetual annuity. In both cases, the ratio must be less than 1 in absolute value for a finite total to exist.
About The Arithmetic vs Geometric Sequence Distinction
A geometric sequence multiplies by a constant ratio. An arithmetic sequence adds a constant difference. The two families of sequences behave differently: arithmetic sequences grow linearly, geometric sequences grow (or decay) exponentially. Do not confuse the formulas. For an arithmetic sequence you use aₙ = a₁ + (n-1)d; for a geometric sequence you use aₙ = a₁ × r^(n-1). The calculator handles only geometric sequences.
| Common Ratio (r) | Term Behaviour | Sum To Infinity |
|---|---|---|
| r > 1 | Terms grow without bound in magnitude | Diverges (no finite sum) |
| r = 1 | All terms equal to a₁ (constant sequence) | Diverges (sum = n × a₁, increases without bound) |
| 0 < r < 1 | Terms approach zero as n increases | Converges to S∞ = a₁/(1 - r) |
| r = 0 | First term is a₁, all subsequent terms are 0 | Converges to S∞ = a₁ |
| -1 < r < 0 | Terms alternate in sign and approach zero | Converges to S∞ = a₁/(1 - r) |
| r = -1 | Sequence alternates between a₁ and -a₁ | Diverges (oscillates between two values) |
| r < -1 | Terms alternate in sign and grow in magnitude | Diverges (no finite sum) |
Common Failure Modes And How To Avoid Them
Sign Error When Finding r From Two Terms
If you are given two non-consecutive terms, the relation r = (aₙ / aₘ)^(1/(n-m)) works only if the result is real. When n - m is even and the ratio of the terms is negative, the root is undefined in real numbers. In that case, the common ratio must be negative, and you need to take the absolute value of the ratio first, then assign the negative sign.
Logarithm Of A Negative Number
When solving for n from a partial sum, the argument of the logarithm can become negative. This happens when a₁ and Sₙ × (1 - r) have opposite signs. The equation then has no real solution, meaning no integer n produces that sum. The calculator will return an error message.
Using The Wrong Relation For r = 1
The standard partial sum relation Sₙ = a₁ × (1 - rⁿ) / (1 - r) is undefined at r = 1 because of division by zero. The correct relation for r = 1 is Sₙ = n × a₁. The calculator switches to this relation automatically, but if you are working by hand, you must remember the special case.
Forgetting To Check Convergence Before Computing S∞
The most common error with infinite geometric series is applying S∞ = a₁ / (1 - r) without first verifying that |r| < 1. The relation returns a finite number even for divergent series if you plug in the numbers, but that number is meaningless. The calculator checks the condition and rejects the calculation when |r| ≥ 1.
Real-World Applications And Who Should Use This Tool
This calculator suits Algebra 2 or precalculus students who need to find terms, the common ratio, or the number of terms from given information. It also serves calculus students who must determine whether an infinite geometric series converges and, if so, compute its sum. Teachers can use it as a reliable answer key or a worked-example generator in class.
Finance
Compound interest, investment growth, and depreciation all follow geometric patterns. The sum of a geometric series gives the total value of an annuity or the total depreciation over time.
Population Biology
Population growth models, cell division, and bacterial reproduction often assume exponential growth, which is a geometric sequence with r > 1.
Physics
Radioactive decay, sound intensity attenuation, and light absorption through a medium all involve repeated multiplication by a factor less than 1, making them geometric sequences.
Chemistry
Half-life calculations and dilution series use the geometric sequence framework to predict concentrations after multiple steps.
An Honest Caveat About The Sum To Infinity
The sum to infinity relation S∞ = a₁ / (1 - r) gives the limit of the partial sums as n goes to infinity. It is not the sum of an infinite number of terms in the ordinary arithmetic sense, you cannot actually add infinitely many numbers. It is the number that the partial sums get arbitrarily close to. For a series that converges, the difference between the partial sum after n terms and the infinite sum becomes smaller than any positive number you choose, provided n is large enough. That is what mathematicians mean by 'sum to infinity'. The calculator computes this limit, and if you need the sum of a specific finite number of terms, use the 'Sum of First n Terms' mode instead.
Common Questions
What is the common ratio (r) in a geometric sequence?
The common ratio is the constant factor between consecutive terms. You find it by dividing any term by the preceding term: r = aₙ / aₙ₋₁. It determines the behaviour of the sequence, growth if |r| > 1, decay if |r| < 1, alternation if r is negative.
What happens if the common ratio is negative?
A negative common ratio causes the terms to alternate in sign. For example, a sequence with a₁ = 5 and r = -2 produces 5, -10, 20, -40, 80, ... The absolute values may grow or shrink depending on |r|, but the sign flips at each step.
What happens when r = 1?
When r = 1, every term equals the first term. The sequence is constant: a₁, a₁, a₁, ... The partial sum relation Sₙ = a₁ × (1 - rⁿ) / (1 - r) is undefined at r = 1 because the denominator is zero. The correct relation is Sₙ = n × a₁. The calculator handles this automatically.
When does the sum to infinity exist?
The sum to infinity of a geometric series exists only when |r| < 1. Under that condition, the terms approach zero as n increases, and the partial sums converge to S∞ = a₁ / (1 - r). When |r| ≥ 1, the series diverges and has no finite sum.
Can I use this calculator for arithmetic sequences?
No. This calculator works only for geometric sequences, where each term is multiplied by a constant ratio. For arithmetic sequences, where each term is added to a constant difference, you need an arithmetic sequence calculator or arithmetic series calculator.
