Arithmetic and geometric sequences

nth termseries sumgeometric seriessum to infinity

The nth term and the sum for both arithmetic and geometric progressions, including the infinite sum that converges.

Arithmetic Sequence nth Term

a_n = a_1 + (n - 1) d

The nth term of a sequence that steps by a fixed common difference.

Arithmetic Series Sum

S_n = \frac{n}{2} \left( 2 a_1 + (n - 1) d \right)

Sum of the first n terms of an arithmetic sequence.

Geometric Sequence nth Term

a_n = a_1 \, r^{\,n-1}

The nth term of a sequence that multiplies by a fixed ratio each step.

Geometric Series Sum

S_n = \frac{a_1 (1 - r^{n})}{1 - r}

Sum of the first n terms of a geometric sequence.

Infinite Geometric Series

S = \frac{a_1}{1 - r}

Limit of a geometric series with ratio between −1 and 1, summed forever.

How they fit together

An arithmetic sequence adds a fixed amount each step; a geometric one multiplies by a fixed ratio. That single difference drives everything else. Arithmetic growth is linear and its sums grow with the square of the number of terms; geometric growth is exponential and its sums are dominated by the last few terms.

The infinite geometric sum is where intuition fails. An endless list of positive numbers can add to a finite total, but only when |r| < 1 — and the formula returns a confident wrong answer if you feed it a ratio outside that range, because nothing in the arithmetic complains. Check the ratio before trusting the sum. This is the resolution of Zeno's dichotomy: the infinitely many halves of a journey add to exactly one journey.