Geometric Sum Calculator

First term: 3, Ratio: 2, Terms: 5 Series: 3, 6, 12, 24, 48 Step 1: S₅ = 3(1 − 2⁵) / (1 − 2) Step 2: S₅ = 3(1 − 32) / (−1) Step 3: S₅ = 3 × 31 = 93

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of such a series, whether finite or infinite, has wide-ranging applications in mathematics, finance, and various scientific fields. Calculating the finite sum involves using a specific formula that accounts for the first term, the common ratio, and the total number of terms. This sum represents the total value accumulated after a certain number of steps in the series. For an infinite geometric series, the sum does not always exist. It only converges to a finite value if the absolute value of its common ratio is less than one. When this condition, |r| < 1, is met, the terms of the series gradually become smaller and smaller, approaching zero, allowing the entire infinite series to sum up to a specific number. If |r| is greater than or equal to one, the terms either grow larger or stay the same size, causing the sum to diverge to infinity, meaning it has no finite sum. Understanding these concepts is crucial for solving problems related to compound interest, depreciation, and even the decay of radioactive materials.