∑ Sequence & Series Calculator
Calculate arithmetic and geometric sequences, find nth terms, and compute series sums
Arithmetic Sequence
aₙ = a₁ + (n-1)d
Arithmetic Series Sum
Sₙ = n/2 · (2a₁ + (n-1)d)
Geometric Sequence
aₙ = a₁ · r^(n-1)
Geometric Series Sum
Sₙ = a₁(1-rⁿ)/(1-r) or S∞ = a₁/(1-r) if |r|<1
Sequence & Series Formulas
Arithmetic Sequence
Geometric Sequence
How to Use This Calculator
- Arithmetic Sequence: Enter the first term (a₁), common difference (d), and term number (n) to find any term in the sequence
- Arithmetic Series: Enter first term, common difference, and number of terms to find the sum of that many terms
- Geometric Sequence: Enter the first term (a₁), common ratio (r), and term number (n) to find any term
- Geometric Series: Enter first term and common ratio. Add number of terms for finite sum, or check infinite sum if |r| < 1
- The calculator displays the first few terms of each sequence to help verify the pattern
- For geometric series, convergence is automatically checked based on the common ratio
- All results update in real-time as you modify any input value
Understanding Sequences & Series
Sequences vs. Series
A sequence is an ordered list of numbers following a specific pattern: a₁, a₂, a₃, a₄, ... Each number is called a term. Sequences can be finite (stopping after n terms) or infinite (continuing forever). A series is the sum of the terms in a sequence: a₁ + a₂ + a₃ + a₄ + ... Written using sigma notation: Σ. While sequences describe patterns, series describe accumulated totals. For example, the sequence 1, 2, 3, 4, ... describes a pattern, while the series 1 + 2 + 3 + 4 + ... = 10 (for first 4 terms) gives a sum. Understanding this distinction is crucial for calculus and analysis.
Arithmetic Sequences & Series
In an arithmetic sequence, each term is obtained by adding a constant difference (d) to the previous term: a, a+d, a+2d, a+3d, ... The formula aₙ = a₁ + (n-1)d lets you find any term without listing all previous terms. Arithmetic sequences have a linear pattern and appear in many contexts: seat numbers in a theater, payment plans with equal installments, temperatures changing at constant rates. To find the sum of an arithmetic series, use Sₙ = n/2 · (first + last) or Sₙ = n/2 · (2a₁ + (n-1)d). This formula comes from pairing terms: (a₁ + aₙ) + (a₂ + aₙ₋₁) + ... each pair sums to the same value.
Geometric Sequences & Series
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio (r): a, ar, ar², ar³, ... The formula aₙ = a₁ · r^(n-1) provides direct access to any term. Geometric sequences model exponential growth and decay: population growth, compound interest, radioactive decay, virus spread. When r > 1, the sequence grows exponentially. When 0 < r < 1, it decays toward zero. When r < 0, terms alternate signs. The sum of a finite geometric series is Sₙ = a₁(1-rⁿ)/(1-r). For infinite geometric series, if |r| < 1, the series converges to S = a₁/(1-r); otherwise it diverges. This convergence formula is powerful for converting repeating decimals to fractions and solving recursive problems.
Convergence & Divergence
An infinite series converges if its partial sums approach a finite limit as n→∞. If not, it diverges. Geometric series: Converges if |r| < 1 with sum a₁/(1-r). Diverges if |r| ≥ 1. Arithmetic series: Always diverge (unless all terms are 0) because adding a constant repeatedly grows without bound. Harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges despite terms approaching 0. Testing convergence requires various methods: ratio test (compare |aₙ₊₁/aₙ|), comparison test (compare to known series), integral test, alternating series test. Convergent series have finite sums and are essential in calculus for Taylor series, Fourier series, and representing functions.
Tips for Working with Sequences & Series
1) Find the pattern: Look at differences (arithmetic) or ratios (geometric) between consecutive terms. 2) Write the first few terms: This helps verify your formula and pattern recognition. 3) Check convergence for infinite series: For geometric, test |r| < 1. For others, use appropriate convergence tests. 4) Use sigma notation: Σᵢ₌₁ⁿ provides compact representation of series. 5) Practice both directions: Find terms from formulas, and find formulas from terms. 6) Connect to real world: Sequences model patterns in nature, finance, physics, and computer science. 7) Memorize key formulas: nth term and sum formulas for arithmetic and geometric sequences are fundamental. 8) Use recursion: Sometimes defining aₙ in terms of aₙ₋₁ is clearer than explicit formulas.
Frequently Asked Questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers (terms): a₁, a₂, a₃, ... For example, 2, 4, 6, 8, ... A series is the sum of the terms of a sequence: a₁ + a₂ + a₃ + ... For example, 2 + 4 + 6 + 8 + ... Sequences describe patterns; series describe accumulated totals. Sequences can be finite or infinite; series can converge (approach a finite sum) or diverge (grow without bound).
What is an arithmetic sequence?
An arithmetic sequence has a constant difference (d) between consecutive terms. Formula: aₙ = a₁ + (n-1)d, where a₁ is the first term, d is common difference, and n is term number. Example: 3, 7, 11, 15, ... (d = 4). The sum of first n terms: Sₙ = n/2 · (a₁ + aₙ) or Sₙ = n/2 · (2a₁ + (n-1)d). Arithmetic sequences appear in many real-world contexts like seat numbering, payment plans, and evenly-spaced intervals.
What is a geometric sequence?
A geometric sequence has a constant ratio (r) between consecutive terms. Formula: aₙ = a₁ · r^(n-1), where a₁ is the first term and r is common ratio. Example: 2, 6, 18, 54, ... (r = 3). The sum of first n terms: Sₙ = a₁(1-rⁿ)/(1-r) for r≠1. For infinite series with |r|<1: S = a₁/(1-r). Geometric sequences model exponential growth/decay like population growth, compound interest, and radioactive decay.
When does an infinite series converge?
For geometric series: converges if |r| < 1 (common ratio's absolute value is less than 1), sum = a₁/(1-r). Diverges if |r| ≥ 1. Arithmetic series never converge unless all terms are 0. Other series require specific tests: ratio test (compare consecutive term ratios), comparison test (compare to known series), integral test, or alternating series test. Convergence means the partial sums approach a finite limit as n→∞.
How do you find the sum of an arithmetic series?
Use the formula Sₙ = n/2 · (first term + last term) or Sₙ = n/2 · (2a₁ + (n-1)d). Example: Sum of first 100 positive integers = 100/2 · (1 + 100) = 50 · 101 = 5050. This formula works because pairing first and last terms, second and second-to-last, etc., gives constant sums. You can also think of it as average of first and last term, multiplied by number of terms.
Is this sequence calculator free to use?
Yes! This sequence and series calculator is completely free with no hidden costs, subscriptions, or limitations. Use it for homework, calculus courses, or learning mathematical sequences.
Is my data private?
Absolutely. All processing happens locally in your browser. Your data never leaves your device and is not stored on any server.