Number sequences,
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Theory & formulas

Arithmetic and geometric sequences, explained.

A sequence is an ordered list of numbers that follows a specific rule. The two most common types — arithmetic and geometric, appearing constantly in finance, physics, and computer science. Both can be fully described by just two numbers: the starting value and a single parameter that controls how each term relates to the previous one.

Arithmetic sequences

In an arithmetic sequence every consecutive pair of terms differs by the same constant amount, called the common difference (d). If the first term is a₁ and you add d repeatedly, you get:

The general term (the formula for any position) and the sum of the first n terms are:

The sum formula has an elegant geometric interpretation: the average of the first and last term, multiplied by the number of terms. This is the method the 10-year-old Gauss reportedly used to sum 1 + 2 + … + 100 in seconds.

Geometric sequences

In a geometric sequence each term is obtained by multiplying the previous term by a fixed number, the common ratio (r). Starting from a₁:

The n-th term and the partial sum are:

When |r| < 1, as n approaches infinity the sum converges to a finite limit: S∞ = a₁ / (1 − r). This is the sum of an infinite geometric series, and it shows up everywhere from Zeno's paradox to compound growth discounting.

Worked example: arithmetic

Find the 12th term and the sum of the first 12 terms of the sequence 5, 8, 11, 14, … Here a₁ = 5 and d = 3.

• a₁₂ = 5 + (12 − 1) · 3 = 5 + 33 = 38
• S₁₂ = 12/2 · (2 · 5 + 11 · 3) = 6 · (10 + 33) = 258

Worked example: geometric

Find the 6th term and sum of 2, 6, 18, 54, … Here a₁ = 2 and r = 3.

• a₆ = 2 · 3⁵ = 2 · 243 = 486
• S₆ = 2 · (1 − 3⁶) / (1 − 3) = 2 · (1 − 729) / (−2) = 728

Special cases to watch for

• d = 0 (arithmetic): every term equals a₁. The sequence is constant and Sₙ = n · a₁.
• r = 1 (geometric): same as above; the sum formula reduces to Sₙ = n · a₁.
• r = 0 (geometric): only the first term is non-zero; every subsequent term is 0.
• r = −1 (geometric): terms alternate between a₁ and −a₁. The partial sum oscillates between a₁ and 0.
• Negative d (arithmetic): the sequence decreases. The terms will eventually cross zero and become negative.
• |r| > 1 (geometric): terms grow without bound. For large n the values can exceed what floating-point arithmetic can represent exactly, so this calculator flags overflow when terms exceed 10¹⁵.

Applications

Arithmetic sequences model anything that grows or shrinks by a fixed amount: straight-line depreciation, salary scales, evenly-spaced sensor readings, stair steps. Geometric sequences describe exponential growth and decay: compound interest, radioactive decay, population doubling, pixel-doubling in a zoom stack.

Disclaimer

All calculations use 64-bit IEEE 754 floating-point. For sequences with very large terms or many decimal places the result is accurate to approximately 15 significant digits. For exact rational arithmetic (e.g. exam proof-writing) work by hand using the formulas above.