Arithmetic Sequence Calculator

The nth term is a_n = a_1 + (n - 1) x d, where a_1 is the first term, d is the common difference, and n is the position in the sequence. Each term equals the one before it plus d. For example, with a_1 = 3 and d = 4, the 5th term is 3 + (5 - 1) x 4 = 19.

An arithmetic sequence adds a fixed amount (d) to each term. A geometric sequence multiplies each term by a fixed ratio (r). For example, 2, 5, 8, 11 is arithmetic with d = 3, while 2, 6, 18, 54 is geometric with r = 3. The growth patterns are linear versus exponential.

Anywhere a quantity grows by a fixed step: simple interest paid on a flat principal, stadium rows with the same number of extra seats per row, monthly savings deposits of the same amount, straight-line depreciation that reduces an asset by the same dollar amount each year, and evenly spaced floor numbers in a building.

Write the sum S_n = a_1 + a_2 + ... + a_n forward, then write it again in reverse. Add the two rows term by term: every paired column sums to (a_1 + a_n), and there are n such pairs, so 2 x S_n = n x (a_1 + a_n). Divide by 2 to get S_n = (n / 2) x (a_1 + a_n).

No. For any non-zero d, the terms grow without bound (positively or negatively), so the partial sums diverge. The only exception is the trivial case where every term is 0 (a_1 = 0 and d = 0), which gives a sum of 0. This calculator works with finite n only.