Arithmetic-Geometric Mean — Definition, Formula & Examples

The arithmetic-geometric mean (AGM) of two positive numbers is the common limit reached by repeatedly taking their arithmetic mean and geometric mean in tandem. Both sequences converge rapidly to a single value that lies between the original geometric and arithmetic means.

Given two positive real numbers $a_0$ and $b_0$ , define the sequences $a_{n+1} = \frac{a_n + b_n}{2}$ and $b_{n+1} = \sqrt{a_n \, b_n}$ . These sequences converge to a common limit $M(a_0, b_0)$ , called the arithmetic-geometric mean, satisfying $\min(a_0, b_0) \le M(a_0, b_0) \le \max(a_0, b_0)$ .

Key Formula

a_{n+1} = \frac{a_n + b_n}{2}, \qquad b_{n+1} = \sqrt{a_n \, b_n}

Where:

$a_n$ = The arithmetic-mean sequence value at step n
$b_n$ = The geometric-mean sequence value at step n
$M(a_0, b_0)$ = The common limit of both sequences (the AGM)

How It Works

Start with two positive numbers

a_0

and

b_0

. At each step, replace them with their arithmetic mean and geometric mean. The arithmetic mean is always at least as large as the geometric mean (by the AM-GM inequality), so the two sequences squeeze together. Convergence is quadratic, meaning the number of correct digits roughly doubles with each iteration. After just a few steps, the two values agree to many decimal places.

Worked Example

Problem: Find the arithmetic-geometric mean of 1 and 4, iterating three times.

Step 1 (n = 0 → 1): Compute the arithmetic and geometric means of 1 and 4.

a_1 = \frac{1 + 4}{2} = 2.5, \qquad b_1 = \sqrt{1 \cdot 4} = 2

Step 2 (n = 1 → 2): Repeat with 2.5 and 2.

a_2 = \frac{2.5 + 2}{2} = 2.25, \qquad b_2 = \sqrt{2.5 \cdot 2} \approx 2.23607

Step 3 (n = 2 → 3): Repeat with 2.25 and 2.23607.

a_3 = \frac{2.25 + 2.23607}{2} \approx 2.24304, \qquad b_3 = \sqrt{2.25 \times 2.23607} \approx 2.24296

Answer: After just three iterations, both sequences agree to about four decimal places:

M(1, 4) \approx 2.2430

. The exact AGM to more digits is approximately 2.24300…

Why It Matters

The AGM appears in Gauss's formula for computing complete elliptic integrals, which arise in orbital mechanics, pendulum periods, and advanced calculus. It also provides one of the fastest known algorithms for computing

\pi

to billions of digits (the Brent–Salamin algorithm).

Common Mistakes

Mistake: Updating both sequences using the new values within the same step (e.g., computing

b_{n+1}

using

a_{n+1}

instead of

a_n

Correction: Both

a_{n+1}

and

b_{n+1}

must be computed from the same pair

(a_n, b_n)

before moving to the next step.