Number Sequence Calculator — Find Patterns & nth Term

How Number Sequences Work

A number sequence is an ordered list of numbers that follows a specific mathematical rule or pattern, where each number is called a term. According to the Encyclopaedia Britannica, sequences have been studied since ancient times, with Euclid's Elements (circa 300 BCE) containing propositions about geometric progressions. The study of sequences and series is a cornerstone of calculus and discrete mathematics, forming the foundation for concepts from convergence to algorithm analysis.

This calculator identifies the two most fundamental sequence types -- arithmetic and geometric -- from three input terms. Arithmetic sequences have a constant difference between consecutive terms, producing linear growth. Geometric sequences have a constant ratio, producing exponential growth or decay. As described by the Wolfram MathWorld reference, the distinction between these sequence types is central to fields ranging from financial mathematics (where annuities follow arithmetic patterns and compound interest follows geometric patterns) to physics (where uniformly accelerated motion follows arithmetic distance increments). The Online Encyclopedia of Integer Sequences (OEIS) catalogs over 370,000 known integer sequences as of 2025, demonstrating the breadth and importance of sequence mathematics.

The Sequence Formulas

For an arithmetic sequence with first term a_1 and common difference d:

nth term: a_n = a_1 + (n - 1) x d

Sum of first n terms: S_n = n x (a_1 + a_n) / 2

For a geometric sequence with first term a_1 and common ratio r:

nth term: a_n = a_1 x r^(n-1)

Sum of first n terms: S_n = a_1 x (1 - r^n) / (1 - r) (when r is not 1)

Worked example: Given the sequence 3, 9, 27, the ratio between consecutive terms is 9/3 = 3 and 27/9 = 3. This is geometric with a_1 = 3, r = 3. The 8th term is 3 x 3^7 = 3 x 2187 = 6,561. The sum of the first 8 terms is 3 x (1 - 3^8) / (1 - 3) = 3 x (-6560) / (-2) = 9,840.

Key Terms You Should Know

Common Difference (d): The constant value added to each term to get the next term in an arithmetic sequence. Calculated as d = a_2 - a_1. A positive d means the sequence is increasing; negative d means decreasing.

Common Ratio (r): The constant multiplier applied to each term to get the next term in a geometric sequence. Calculated as r = a_2 / a_1. When |r| > 1, terms grow exponentially; when |r| < 1, terms shrink toward zero.

Convergent Series: An infinite series whose partial sums approach a finite limit. A geometric series converges only when |r| < 1, with the sum S = a_1 / (1 - r).

Divergent Series: An infinite series whose partial sums grow without bound. Arithmetic series with d not equal to 0 always diverge, as do geometric series with |r| >= 1.

Recursive Formula: A definition that expresses each term in relation to previous terms. For arithmetic: a_n = a_(n-1) + d. For geometric: a_n = a_(n-1) x r. The Fibonacci sequence uses a_n = a_(n-1) + a_(n-2).

Common Number Sequences Reference

The table below shows well-known number sequences, their types, and formulas. These sequences appear frequently in mathematics courses and standardized tests.

Practical Examples

Example 1 -- Saving for retirement: You deposit $500 each month into a savings account. After 1 month you have $500, after 2 months $1,000, after 3 months $1,500. This is an arithmetic sequence with a_1 = 500 and d = 500. After 30 years (360 months), the total deposited is S_360 = 360 x (500 + 180,000) / 2 = $64,980,000... wait, the nth term is a_360 = 500 + 359 x 500 = $180,000. S_360 = 360 x (500 + 180,000) / 2 = $32,490,000? Actually, the cumulative total is simply n x a_1 = 360 x 500 = $180,000. The sum formula applies when summing the sequence terms themselves. Use our compound interest calculator to factor in interest growth.

Example 2 -- Bacterial growth: A bacterial colony doubles every hour, starting with 100 cells. The sequence is 100, 200, 400, 800... This is geometric with a_1 = 100 and r = 2. After 24 hours, the count is 100 x 2^23 = 838,860,800, nearly 839 million cells. The total cells produced over 24 hours is S_24 = 100 x (1 - 2^24) / (1 - 2) = 1,677,721,500.

Example 3 -- Depreciation schedule: A $30,000 car depreciates by $3,000 each year. The values form an arithmetic sequence: 30000, 27000, 24000, 21000... with d = -3000. After 8 years, the value is 30000 + 7 x (-3000) = $9,000. Use the logarithm calculator to solve for the time needed to reach a specific value in geometric depreciation models.

Tips and Strategies

• Check differences first: Compute the differences between consecutive terms. If they are constant, the sequence is arithmetic. If not, check if the ratios are constant for a geometric sequence.
• Use second differences for quadratic sequences: If the first differences are not constant but the second differences are, the sequence is quadratic (polynomial of degree 2). The percentage calculator can help verify growth rates between terms.
• Test convergence before summing infinitely: For geometric series, check that |r| < 1 before applying the infinite sum formula S = a_1 / (1 - r). Otherwise, the sum does not exist.
• Use the explicit formula for large n values: Computing the 1000th term by iterating through all 999 previous terms is inefficient. The explicit formula a_n = a_1 + (n-1)d or a_n = a_1 x r^(n-1) gives the answer directly.
• Relate sequences to real-world models: Arithmetic = linear growth (salary raises), geometric = exponential growth (investments), quadratic = accelerating growth (falling objects). Identifying the model helps predict future values.
• Verify with multiple terms: Three terms can suggest a pattern, but verifying with at least four or five terms increases confidence. A sequence starting 1, 2, 4 could be geometric (r=2) or the beginning of 1, 2, 4, 7, 11 (differences increasing by 1).