# Geometric Series

## Definition

• A Geometric Series is a sequence of numbers in which the ratio of any two consecutive terms is constant. This ratio is also known as the common ratio.
• The series is typically expressed as: a, ar, ar^2, ar^3, …, where ‘a’ is the first term and ‘r’ is the common ratio.

## Formula

• The nth term of a Geometric Series, denoted as u_n, can be represented as u_n = ar^(n-1).
• The sum to infinity of a Geometric Series, denoted as S_∞, can be given by S_∞ = a / (1-r), given that the absolute value of ‘r’ is less than 1.
• The sum of the first n terms of a Geometric Series, denoted as S_n, can be calculated by S_n = a(1 - r^n) / (1 - r), for r ≠ 1.

## Properties

• The common ratio ‘r’ in a Geometric Series can be calculated by dividing any term by the preceding term in the series.
• If the common ratio is positive, the terms in the Geometric Series will alternate between positive and negative.

## Techniques & Problems

• To find the common ratio ‘r’, divide any term of the sequence by the term before it.
• To determine whether a sequence is geometric, check if the common ratio is consistent across all consecutive pairs of terms.
• Problems involving Geometric Series frequently require utilising the formulae to identify a specific term, compute the sum of several terms, or solve for the common ratio.

## Applications

• Geometric Series have numerous applications in many real-world situations such as in calculating population growth, appreciating assets, interest compounding, computer science (e.g., algorithms), and more.
• A thorough understanding of Geometric Series and the ability to apply the formulas are integral aspects of mathematical problem-solving, data analysis, and numerous scientific computations.

## Importance of Practice

• Understanding Geometric Series necessitates regular practice with determining the common ratio, applying the appropriate formula, and understanding the relationship between different terms within the sequence.
• Regular practice with real-life examples can help deepen understanding of the concept and practical applications of Geometric Series.