Problem 65 - Algebra and Trigonometry

Question

Write the sum using sigma notation. $$1+x+x^{2}+x^{3}+\cdots+x^{100}$$

Step-by-Step Solution

Answer

\(\sum_{n=0}^{100} x^n\)

Step 1: Identify the Pattern

The given series is a geometric progression: \(1 + x + x^2 + x^3 + \cdots + x^{100}\). The general term for the sequence can be determined as \(x^n\), where \(n\) starts from 0 and goes up to 100.

Step 2: Determine the Structure of Sigma Notation

In sigma notation, the sum of the series is expressed as \(\sum_{n=a}^{b} f(n)\), where \(a\) is the starting index, \(b\) is the ending index, and \(f(n)\) is the general term of the sequence.

Step 3: Apply Sigma Notation to the Series

Using the information determined from the sequence, the series can be written in sigma notation as \(\sum_{n=0}^{100} x^n\). Here, \(n\) represents the index that goes from 0 to 100, and \(x^n\) is the general term of the series.

Key Concepts

Geometric Series

A geometric series is a sum of the terms in a geometric sequence. In such sequences, each term is obtained by multiplying the previous term by a constant called the "common ratio". The series given in the exercise, \(1 + x + x^2 + x^3 + \cdots + x^{100}\), is an example of a geometric series. In this series, each term is a power of \(x\). The first term is 1, the common ratio is \(x\), and the series continues until \(x^{100}\).
Geometric series are important because they allow us to simplify complex expressions and solve problems related to them. They are widely used in mathematics, physics, economics, and computer science to model growth, decay, and compound interest scenarios.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers and operations in mathematics. It helps simplify the communication of complex mathematical concepts and calculations. In the context of the provided exercise, sigma notation is a specific type of mathematical notation used to represent series.
Specifically, sigma notation \(\sum\) allows us to express the sum of a series in a concise form. In our exercise, \(\sum_{n=0}^{100} x^n\) is used to represent the sum from \(x^0\) to \(x^{100}\). The Greek letter "\(\Sigma\)" denotes summation. The expression below it, such as \(n=0\), indicates the starting index, while the expression on top, \(100\), indicates the ending index. These indices tell us the range over which the summation occurs, while the main expression, \(x^n\), describes the terms we sum.

Sequences and Series

Understanding sequences and series is foundational in mathematics. A sequence is an ordered list of numbers, and each number in the list is called a term. When these terms are added together, we form a series.
In sequences, each term can often be described by a formula, which lets us determine any term's value without listing all preceding terms. In the series \(1 + x + x^2 + x^3 + \cdots + x^{100}\), the sequence's general term is \(x^n\).
Series, such as arithmetic and geometric, are built upon sequences. Arithmetic series have a constant difference between terms, while geometric series have a constant ratio, as seen in our example. Recognizing and writing series in concise forms like sigma notation is crucial for mathematical problem-solving and analysis.