Arithmetic Sequence
aₙ = a₁ + (n − 1) × d
Sₙ = n/2 × (2a₁ + (n − 1)d)
Geometric Sequence
aₙ = a₁ × r^(n − 1)
Sₙ = a₁ × (1 − rⁿ) / (1 − r)
S∞ = a₁ / (1 − r), |r| < 1
Arithmetic: 2, 5, 8, 11, ...
a₁ = 2, d = 3
Geometric: 3, 6, 12, 24, ...
a₁ = 3, r = 2
Convergent: 1, 0.5, 0.25, ...
a₁ = 1, r = 0.5, S∞ = 2
A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the sequence is called a term. The two most common types are arithmetic sequences (where each term differs from the previous by a constant amount) and geometric sequences (where each term is multiplied by a constant factor from the previous term).
A series is the sum of the terms in a sequence. When we add up a finite number of terms, we get a finite series. Some geometric series with a common ratio between -1 and 1 (exclusive) can be summed infinitely, converging to a specific value called the infinite sum.
1. Select the sequence type: Arithmetic (constant difference between terms) or Geometric (constant ratio between terms).
2. Enter the first term (a₁) of your sequence.
3. Enter the common difference (d) for arithmetic or common ratio (r) for geometric sequences.
4. Enter the term number (n) to find the n-th term.
5. Optionally, specify how many terms to sum. Toggle step-by-step solutions and sequence list for detailed output.
Disclaimer
Sequence and series calculations follow standard mathematical formulas. Results depend on correct input values and assumptions. For very large terms or ratios, numerical precision may be affected.