Geometric Series

= ·

Tap a term to see what it means, or .

Calculate

Example: to cross a room, you walk half the way, then half of what is left, and so on. After 10 stages, how much of the room have you covered? And if the stages never ended? Change the values to work out your own case.

First term a Ratio r Number of terms n

Step by step

What each term means

Sum of the First n Terms (Sₙ): The total of a + ar + ar² + … + arⁿ⁻¹: n terms, each one the previous multiplied by the ratio r.
First Term (a): The term that opens the sequence. All the others are it multiplied by powers of r.
Numerator (1 − rⁿ): It comes from a trick: multiplying the sum by r, almost every term repeats, and the difference Sₙ − rSₙ leaves only a − arⁿ. When |r| < 1, rⁿ goes to zero and the numerator becomes 1.
Denominator (1 − r): It completes the trick: Sₙ(1 − r) = a(1 − rⁿ). With r = 1 the formula doesn't hold, and the sum is simply n · a. With |r| < 1, the infinite sum is a / (1 − r).