Calculating the Sum of the Geometric Series: A Comprehensive Guide

Understanding how to calculate the sum of a geometric series is a fundamental skill in mathematics. This guide will walk you through the process of calculating the sum of the series An dfrac{2^n}{2}. Specifically, we will focus on the series Sn A0 A3 A6 ... A3n.

This series can be expressed as:

Sn dfrac{2^0}{2} dfrac{2^3}{2} dfrac{2^6}{2} ... dfrac{2^{3n}}{2}

Let's simplify this series step by step:

First, we can rewrite each term in the series:

Ak dfrac{2^{3k}}{2}

Thus, the series Sn can be written as:

Sn dfrac{2^0}{2} dfrac{2^3}{2} dfrac{2^6}{2} ... dfrac{2^{3n}}{2}

Transforming the Equation

To simplify the expression, let's define a new sequence:

bn 2^{3n-1}

Mapping the Sequences

We can express bn in terms of a3n, where:

an dfrac{2^n}{2} 2^{n-1}

Therefore:

bn a3n

The series Sn can now be expressed as:

Sn b0 b1 b2 ... bn

Substitute the expression for bn in the series:

Sn 2^{0} 2^{4} 2^{7} ... 2^{3n-1}

This can be simplified to:

Sn sum_{k0}^{n} 2^{3k}

Converting to a Geometric Series

We can further convert this sum into a standard form of a geometric series:

sum_{k0}^{n} 2^{3k} dfrac{1}{2} sum_{k0}^{n} 2^{3k} dfrac{1}{2} sum_{k0}^{n} (2^3)^k

This is a geometric series with the first term A 1, common ratio R 2^3, and number of terms N n 1. The sum of the first N terms of a geometric series is given by:

SN A left[ dfrac{R^N - 1}{R - 1} right]

Therefore:

sum_{k0}^{n} 8^k dfrac{1 - 8^{n 1}}{1 - 8} dfrac{8^{n 1} - 1}{7}

Final Calculation

Substitute this back into the expression for Sn:

Sn dfrac{1}{2} times dfrac{8^{n 1} - 1}{7} dfrac{8^{n 1} - 1}{14}

This is the final result for the sum of the given series. The process illustrated here is a valuable approach to solving similar problems involving the sum of geometric series.

Key Takeaways

Understanding the structure of a geometric series is crucial. Transforming the series into a standard form can simplify the calculation process. Using the geometric series sum formula is a powerful tool in solving these types of problems.

Important Formulas

Sum of a geometric series: SN A left[ dfrac{R^N - 1}{R - 1} right] Sn dfrac{8^{n 1} - 1}{14}

This guide provides a clear and detailed explanation to help you master the techniques for calculating the sum of geometric series.