The Sequence Calculator helps you work with mathematical sequences—ordered lists of numbers following specific patterns. It can identify sequence types, find terms, calculate sums, and determine formulas for arithmetic and geometric sequences, as well as more complex patterns.

Key Features

• Sequence Identification: Automatically detect whether a sequence is arithmetic, geometric, or follows another pattern.
• Find nth Term: Calculate any term in the sequence using the general formula.
• Sum Calculation: Compute the sum of the first n terms (arithmetic or geometric series).
• Common Difference/Ratio: Identify the common difference (arithmetic) or common ratio (geometric).
• Formula Generation: Generate explicit and recursive formulas for sequences.
• Pattern Recognition: Analyze Fibonacci sequences, triangular numbers, square numbers, and other special sequences.

Types of Sequences

Arithmetic Sequence: Each term differs from the previous by a constant amount (common difference d).

Geometric Sequence: Each term is found by multiplying the previous by a constant (common ratio r).

Example 1: Arithmetic Sequence

Find the 20th term of the sequence 5, 9, 13, 17, ...:

Step 1: Identify the pattern
9 - 5 = 4
13 - 9 = 4
17 - 13 = 4
Common difference d = 4

Step 2: Use arithmetic sequence formula
a<sub>n</sub> = a<sub>1</sub> + (n - 1)d
a<sub>2</sub><sub>0</sub> = 5 + (20 - 1)(4)
a<sub>2</sub><sub>0</sub> = 5 + 19(4)
a<sub>2</sub><sub>0</sub> = 5 + 76
a<sub>2</sub><sub>0</sub> = 81

The 20th term is 81.

Example 2: Geometric Sequence

Find the 7th term of the sequence 3, 12, 48, 192, ...:

Step 1: Identify the pattern
12 ÷ 3 = 4
48 ÷ 12 = 4
192 ÷ 48 = 4
Common ratio r = 4

Step 2: Use geometric sequence formula
a<sub>n</sub> = a<sub>1</sub> × r⁽<sup>n</sup>⁻<sup>1</sup>⁾
a<sub>7</sub> = 3 × 4⁽<sup>7</sup>⁻<sup>1</sup>⁾
a<sub>7</sub> = 3 × 4<sup>6</sup>
a<sub>7</sub> = 3 × 4096
a<sub>7</sub> = 12,288

The 7th term is 12,288.

Example 3: Arithmetic Series Sum

Find the sum of the first 15 terms of 2, 5, 8, 11, ...:

Given: a<sub>1</sub> = 2, d = 3, n = 15

Method 1: Using the sum formula
S<sub>n</sub> = (n/2)[2a<sub>1</sub> + (n - 1)d]
S<sub>1</sub><sub>5</sub> = (15/2)[2(2) + (15 - 1)(3)]
S<sub>1</sub><sub>5</sub> = (15/2)[4 + 42]
S<sub>1</sub><sub>5</sub> = (15/2)(46)
S<sub>1</sub><sub>5</sub> = 15 × 23 = 345

Method 2: Using first and last term
First find a<sub>1</sub><sub>5</sub>: a<sub>1</sub><sub>5</sub> = 2 + 14(3) = 44
S<sub>n</sub> = (n/2)(a<sub>1</sub> + a<sub>n</sub>)
S<sub>1</sub><sub>5</sub> = (15/2)(2 + 44) = (15/2)(46) = 345

The sum is 345.

Example 4: Geometric Series Sum

Find the sum of the first 6 terms of 2, 6, 18, 54, ...:

Given: a<sub>1</sub> = 2, r = 3, n = 6

Use geometric series formula:
S<sub>n</sub> = a<sub>1</sub>(1 - r<sup>n</sup>)/(1 - r)    [when r ≠ 1]
S<sub>6</sub> = 2(1 - 3<sup>6</sup>)/(1 - 3)
S<sub>6</sub> = 2(1 - 729)/(-2)
S<sub>6</sub> = 2(-728)/(-2)
S<sub>6</sub> = -1456/(-2)
S<sub>6</sub> = 728

The sum is 728.

Example 5: Fibonacci Sequence

Find the 10th Fibonacci number:

Fibonacci sequence: each term is the sum of the previous two
Starting with F<sub>1</sub> = 1, F<sub>2</sub> = 1

F<sub>1</sub> = 1
F<sub>2</sub> = 1
F<sub>3</sub> = 1 + 1 = 2
F<sub>4</sub> = 1 + 2 = 3
F<sub>5</sub> = 2 + 3 = 5
F<sub>6</sub> = 3 + 5 = 8
F<sub>7</sub> = 5 + 8 = 13
F<sub>8</sub> = 8 + 13 = 21
F<sub>9</sub> = 13 + 21 = 34
F<sub>1</sub><sub>0</sub> = 21 + 34 = 55

The 10th Fibonacci number is 55.

• Check Differences: For arithmetic sequences, check if consecutive differences are constant. For geometric, check if consecutive ratios are constant.
• Recursive vs Explicit: Recursive formulas define each term using previous terms (a_n = a_n₋₁ + d). Explicit formulas give any term directly (a_n = a₁ + (n-1)d).
• Geometric Series Convergence: For infinite geometric series, if |r| < 1, the sum converges to a₁/(1-r).
• Negative Common Difference: Arithmetic sequences can decrease. If d < 0, the sequence decreases.
• Zero Common Ratio: If r = 0 in a geometric sequence, all terms after the first are zero.
• Alternating Signs: If r < 0 in a geometric sequence, signs alternate between positive and negative.
• Pattern Recognition: Look for special sequences: squares (1, 4, 9, 16, ...), triangular numbers (1, 3, 6, 10, ...), powers (1, 2, 4, 8, ...).
• Index Notation: Be careful with indexing: some sequences start at n=0, others at n=1. Check which convention is being used.