Geometric Progression: Concepts, Formulas and Examples

A geometric progression, or GP, is a sequence in which each term after the first is found by multiplying the previous term by the same fixed number. That fixed multiplier is called the common ratio and is written r. For example, 3, 6, 12, 24, 48, … is a GP with first term a = 3 and common ratio r = 2, because every step multiplies by 2.

Geometric progressions matter because proportional growth is one of nature's favourite patterns. Bacteria doubling each hour, money compounding each year, a rumour spreading from person to person, and computer memory sizes (1, 2, 4, 8, 16 GB) all follow geometric progressions. Unlike an AP's fixed addition, a GP's fixed multiplication grows far faster.

Key Concepts

• First term (a): the starting value of the sequence.
• Common ratio (r): the fixed multiplier, found by r = T_2 / T_1 = T_3 / T_2, and so on. It is identical for every neighbouring pair.
• General form: a, ar, ar^2, … , where the exponent of r trails the term number by one.
• nth term: T_n = a times r^(n - 1). Reaching the nth term needs (n - 1) multiplications by r.
• Sum of n terms: S_n = a times (r^n - 1)/(r - 1) when r is not 1, and S_n = n times a when r = 1.

Formulas

• nth term: T_n = a times r^(n - 1), for positive integer n.
• Sum of first n terms: S_n = a times (r^n - 1)/(r - 1) for r not equal to 1; S_n = n times a for r = 1.
• Equivalent form: S_n = a times (1 - r^n)/(1 - r), which is often more convenient when r is a fraction below 1.
• Assumptions: a is non-zero and r is constant through the sequence.

Worked Example

Take the GP 3, 6, 12, 24, … with a = 3 and r = 2. The 10th term is T_10 = 3 times 2^9 = 3 times 512 = 1536. The sum of the first 8 terms is S_8 = 3 times (2^8 - 1)/(2 - 1) = 3 times 255 = 765. Check: 3 + 6 + 12 + 24 = 45, and the formula gives 3 times 15 = 45.

Common Misconceptions

• r must be bigger than 1: false. A fractional ratio gives a shrinking GP like 81, 27, 9, 3, … with r = 1/3, and a negative ratio gives alternating signs like 2, -6, 18, -54, … with r = -3. Both are valid GPs.
• GP terms must be whole numbers: no. Ratios like 1.5 or square root of 2 are perfectly fine; the terms just become decimals or irrationals.
• The nth term is a times r^n: the off-by-one error again. Since the first term is already a (a times r^0), the nth term uses r^(n - 1).

Real-World Applications

• Finance: compound interest grows as A = P times (1 + r)^n, a GP with ratio (1 + r).
• Biology: cell division and bacterial colonies double (or multiply) per generation.
• Physics: radioactive decay shrinks the remaining mass by a fixed fraction each half-life.

A geometric progression multiplies by a constant ratio r at every step, described fully by its first term a and that ratio. The nth term T_n = a times r^(n - 1) reaches any position, and S_n = a times (r^n - 1)/(r - 1) totals the terms.