Series Calculator

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Reviewed by: David Chen, CFA

Welcome to the **Geometric Series Calculator**. This tool is essential for mathematicians, engineers, and financial analysts for quickly finding the sum of 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.

Geometric Series Calculator

The Sum of the Series ($S$) is: —

Detailed Calculation Steps

Click 'Calculate' to see the step-by-step process.

series calculator Formula: Geometric Series Sum

The sum of a finite geometric series is calculated using the following formula:

S = a * (1 - r^n) / (1 - r)

Formula Source: Wikipedia – Geometric series | Khan Academy

Variables Explained

  • First Term ($a$): The value of the first term in the series.
  • Common Ratio ($r$): The ratio of any term to the previous term. This value is constant throughout the series.
  • Number of Terms ($n$): The total count of terms being summed in the series.
  • Sum of Series ($S$): The final calculated result, which is the total sum of all terms.

What is a series calculator?

A series calculator, in a general sense, is a tool designed to compute the sum of a sequence of numbers (a series). These calculators are indispensable in fields like finance, physics, and computer science where patterns and sequences are common.

Specifically, this tool focuses on the **Geometric Series**. A geometric series is a sequence where each successive term is found by multiplying the previous one by a fixed number, the common ratio ($r$). For example, 2, 4, 8, 16, 32… is a geometric series with $a=2$ and $r=2$.

Understanding and calculating series is crucial for modeling exponential growth (like compound interest) or decay, and for analyzing the efficiency of algorithms.

How to Calculate Geometric Series Sum (Example)

Let’s calculate the sum of a geometric series where the first term is 3, the common ratio is 4, and there are 5 terms (3, 12, 48, 192, 768).

  1. Identify Variables: Set $a = 3$, $r = 4$, and $n = 5$.
  2. Apply the Formula: The formula is $S = a \cdot \frac{1 – r^n}{1 – r}$.
  3. Calculate the Numerator: $r^n = 4^5 = 1024$. The numerator is $1 – 1024 = -1023$.
  4. Calculate the Denominator: The denominator is $1 – r = 1 – 4 = -3$.
  5. Compute the Sum: $S = 3 \cdot \frac{-1023}{-3} = 3 \cdot 341 = 1023$.

The total sum of the first 5 terms of the series is 1023.

Frequently Asked Questions (FAQ)

What happens if the Common Ratio ($r$) is 1?

If $r=1$, the standard formula is undefined (division by zero). In this special case, every term is the same as the first term, $a$. Therefore, the sum is simply $S = a \cdot n$. Our calculator handles this specific edge case.

What is the difference between a series and a sequence?

A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8…). A series is the sum of the terms of a sequence (e.g., 2 + 4 + 6 + 8…).

Can this calculator find the common ratio ($r$)?

No, this calculator is currently designed to solve for the Sum ($S$) given $a$, $r$, and $n$. Solving for $r$ would require complex numerical methods beyond the scope of a simple single-variable solver.

Is the Geometric Series the same as Compound Interest?

They are fundamentally related. The balance in a compound interest account over time forms a geometric sequence, and the total value of an annuity (a series of payments) is calculated using the sum of a geometric series formula.

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