Find the nth Term and Sum of Any Sequence with Full Working
Enter the parameters for an arithmetic, geometric, or Fibonacci sequence to instantly generate the full sequence, find any nth term, calculate the sum, and see step-by-step formula derivations.
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Number Sequence Calculator
Generate arithmetic, geometric, or Fibonacci sequences, find the nth term and sum, and see the step-by-step formula derivation.
Arithmetic Parameters
Choose a sequence type and enter parameters to see results
Formula Reference
Arithmetic
aₙ = a₁ + (n-1)d
Sₙ = n/2[2a₁ + (n-1)d]
Linear growth
Geometric
aₙ = a₁ x r^(n-1)
Sₙ = a₁(1-rⁿ)/(1-r)
Exponential growth
Fibonacci
F(0)=0, F(1)=1
F(n) = F(n-1) + F(n-2)
phi = (1+sqrt5)/2
Arithmetic Sequences
Enter a first term, common difference, and target position. The calculator generates all terms up to n, finds the nth term using aₙ = a1 + (n-1)d, and computes the sum using the arithmetic series formula Sₙ = n/2 x [2a1 + (n-1)d]. Step-by-step derivation included.
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Geometric Sequences
Enter a first term, common ratio, and position. The calculator generates the sequence, finds the nth term using aₙ = a1 x r^(n-1), and computes the sum using Sₙ = a1(1 - r^n) / (1 - r) (or Sₙ = n x a1 when r = 1). Works with ratios between -10 and 10.
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Fibonacci Sequences
Enter any position up to 100 to generate the full Fibonacci sequence from F(0)=0, F(1)=1 up to F(n). The result shows the nth term, the sum of all terms, and the golden ratio (phi = (1 + sqrt(5)) / 2 approx. 1.618). A step-by-step term derivation is available.
Try It NowCalculate any sequence in three steps
Choose a sequence type
Click the Arithmetic, Geometric, or Fibonacci tab. Each type shows its own input fields: first term and common difference for arithmetic; first term and ratio for geometric; just a position for Fibonacci.
Enter the parameters
Fill in the inputs and click the calculate button. The position (n) determines how many terms are generated and which term is identified as the nth term.
Review results and steps
The result cards show the nth term, sum, and key parameters. Click "Show Calculation Steps" to expand a full formula derivation with substituted values.
Deep Dive
How Each Sequence Type Works
The three sequence types have distinct growth patterns, formulas, and applications.
What This Calculator Provides
3 Sequence Types
Arithmetic, geometric, and Fibonacci in one calculator.
nth Term and Sum
Both key values computed with substituted formulas shown.
Full Sequence Display
First 15-20 terms shown in a scrollable list.
Step-by-Step
Expandable derivation with formula and calculation.
Who Uses It
Who Uses This Calculator
Students, teachers, and anyone working with mathematical sequences.
Frequently Asked Questions
What types of sequences does the calculator support?
The calculator supports three types. Arithmetic sequences: each term adds a constant difference (e.g. 2, 5, 8, 11...). Geometric sequences: each term is multiplied by a constant ratio (e.g. 3, 6, 12, 24...). Fibonacci sequences: each term is the sum of the two preceding terms (0, 1, 1, 2, 3, 5, 8, 13...).
How do I find the nth term of an arithmetic sequence?
Use the formula aₙ = a1 + (n-1) x d, where a1 is the first term, d is the common difference, and n is the position. For example, in the sequence 3, 7, 11, 15 (a1=3, d=4), the 10th term is 3 + (10-1) x 4 = 3 + 36 = 39. The calculator shows this substitution in the step-by-step view.
What is the difference between arithmetic and geometric sequences?
Arithmetic sequences add a constant to each term (linear growth). Geometric sequences multiply by a constant (exponential growth or decay). For example, 2, 5, 8, 11 is arithmetic (adding 3). The sequence 2, 6, 18, 54 is geometric (multiplying by 3). The growth rate is very different: by the 10th term, the arithmetic sequence reaches 29, while the geometric sequence reaches 39,366.
How does the Fibonacci sequence work and what is the golden ratio?
Starting from F(0)=0 and F(1)=1, each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21... The ratio of consecutive terms converges toward the golden ratio phi = (1 + sqrt(5)) / 2 approx. 1.618. For example, 21/13 is approximately 1.615 and 34/21 is approximately 1.619. The golden ratio appears in plant spiral counts, shell shapes, and art proportions.
How is the sum of a sequence calculated?
For arithmetic sequences: Sₙ = n/2 x [2a1 + (n-1)d]. For geometric sequences: Sₙ = a1(1 - r^n) / (1 - r) when r is not 1, or Sₙ = n x a1 when r = 1. For Fibonacci sequences, the sum is computed by adding all generated terms. The calculator shows the full formula with substituted values in the step-by-step view.
What are some real-world applications of these sequences?
Arithmetic sequences: salary raises by fixed amounts, evenly spaced measurements, staircase step heights. Geometric sequences: compound interest growth, population doubling, radioactive decay, loan balance reduction. Fibonacci sequences: petal counts in flowers (3, 5, 8, 13 petals are common Fibonacci numbers), seed spiral counts in sunflowers, and proportional design using the golden ratio.
Ready to Explore Sequences?
Choose a sequence type, enter your parameters, and get the nth term, sum, and full derivation instantly.