Arithmetic and geometric sequences

You put 3 euros in a piggy bank, then 5 more euros every week: 3, 8, 13, 18... A rumour spreads: one person on day one, then each person tells two others: 1, 2, 4, 8, 16... These lists of numbers evolving by a rule are called sequences.

We call a sequence u, and its term number n is written u index n, here noted u(n). The first term is often u(0). A sequence can be defined in two ways: by recurrence (how to go from one term to the next) or by an explicit formula (how to compute term number n directly, without computing all the previous ones).

An arithmetic sequence moves forward by always adding the same number r, called the common difference: u(n+1) = u(n) + r. Its explicit formula is u(n) = u(0) + n × r. The piggy bank is one: u(0) = 3, r = 5, so u(n) = 3 + 5n. To recognise an arithmetic sequence, check that the difference between consecutive terms is constant.

A geometric sequence moves forward by always multiplying by the same number q, its common ratio: v(n+1) = v(n) × q. Its explicit formula is v(n) = v(0) × q to the power n. The rumour is one: v(0) = 1, q = 2. To recognise it, check that the quotient of consecutive terms is constant.

The foolproof test: from one term to the next, if you always ADD the same amount, it is arithmetic; if you always MULTIPLY by the same amount, it is geometric. A 2% yearly increase, for example, multiplies by 1.02 each year: that is geometric, not arithmetic!