Geometric Progression

A geometric progression (GP) is a sequence of numbers in which each term, after the first, is obtained by multiplying the previous term by a fixed non-zero number called the common ratio, written r. For example 2, 6, 18, 54 is a GP because every term is three times the one before it, so r = 3. Unlike an arithmetic progression, where you add a fixed amount each time, in a GP you multiply by a fixed amount. The common ratio is found by dividing any term by the term immediately before it; in a genuine GP that quotient is the same everywhere.

Writing a GP out shows a clear pattern: first term a, second term ar, third term ar², fourth term ar³, and so on. The power of r is always one less than the position of the term, because the first term has not been multiplied yet. This gives the formula for the nth term: Tn = a r^(n−1). With this single formula you can jump straight to, say, the 12th term without listing the first eleven. Knowing any two of a, r and Tn lets you find the third.

Adding the first n terms of a GP gives Sn = a(rⁿ − 1)/(r − 1) when r is greater than 1, and the equivalent form Sn = a(1 − rⁿ)/(1 − r) when r is less than 1; both give the same value, you simply pick the version that keeps the arithmetic positive and tidy. The formula is derived by writing Sn, multiplying it by r, and subtracting, which makes most terms cancel. If r equals exactly 1 every term is the same, so the sum is just n × a.

If three numbers are in GP, the middle one is the geometric mean of the outer two. For numbers a and b the geometric mean is the square root of their product, GM = sqrt(a×b), and inserting it between a and b makes a three-term GP. This is the multiplicative cousin of the arithmetic mean (a+b)/2. Geometric means are used wherever growth is multiplicative, such as average rates of return or population growth.