Number Sequence Calculator
Find the nth term and sum of an arithmetic or geometric sequence.
Assumptions used in this calculation
- Sequence type: The same "common difference / ratio" input is interpreted as an additive difference for arithmetic sequences or a multiplicative ratio for geometric sequences, based on the selected mode.
About this calculator
Finding the 20th term of a sequence by manually adding a common difference nineteen times, or multiplying a common ratio to the 19th power, is slow and compounds any early arithmetic slip through every later term. This calculator finds the nth term directly using the closed-form formulas: aₙ = a₁ + (n−1)d for arithmetic sequences (where consecutive terms differ by a constant additive amount) and aₙ = a₁ × r^(n−1) for geometric sequences (where consecutive terms differ by a constant multiplicative ratio), so no term-by-term iteration is needed. It also computes the sum of the first n terms using Sₙ = n/2 × (2a₁ + (n−1)d) for arithmetic sequences and Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r) for geometric sequences, since summing a long sequence term-by-term is even more error-prone than finding a single term. That turns a repetitive, cumulative calculation into a single formula evaluation.
Worked example
a₁=2, d=3, n=5 (arithmetic)
Result: 5th term = 14, sum of first 5 terms = 40
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