Problem 67 - Algebra and Trigonometry

Question

\(61-68\) Write the sum using sigma notation. $$ 1+x+x^{2}+x^{3}+\dots+x^{100} $$

Step-by-Step Solution

Answer

The sum is written in sigma notation as \(\sum_{n=0}^{100} x^n\).

Step 1: Identify the pattern

The given series is an arithmetic expression with terms following the form of powers of \(x\). The pattern begins at \(x^0\) and continues to \(x^{100}\).

Step 2: Understand sigma notation

Sigma notation is used to succinctly represent the sum of a sequence. It uses the Greek letter \(\Sigma\) to indicate summation and typically includes terms like the starting and ending indices, as well as the general formula for the terms being summed.

Step 3: Determine the index of summation

For this series, we start summing from \(n = 0\) to \(n = 100\). Thus, \(n\) serves as the index of summation corresponding to the powers of \(x\).

Step 4: Write the sigma notation

Using sigma notation, the expression becomes \(\sum_{n=0}^{100} x^n\). This indicates the sum of \(x^n\) where \(n\) ranges from 0 to 100.

Key Concepts

Arithmetic Expression

The term "arithmetic expression" refers to a sequence of numbers or variables combined through operations such as addition, subtraction, multiplication, or division. In our example, we observe an arithmetic expression composed of powers of \(x\) that are added together.

• Each term in this sequence is a power of \(x\), starting from \(x^0\) (which equals 1) and extending through \(x^{100}\).
• The terms are connected by the addition operator, forming a continuous sum.

In context, understanding this sequence helps us visualize the overall structure of the problem, highlighting that each subsequent term increases in complexity as it powers up \(x\).

Summation Index

A summation index is a variable used within sigma notation to control the iteration of terms. For our sum of powers of \(x\), the summation index plays a crucial role in determining which terms are included.

• Here, the index used is \(n\).
• It begins at a starting point, \(n = 0\), and continues through our specified range, up to \(n = 100\).
• This provides a complete identifier for each term in our series as \(x^n\), where \(n\) progressively takes on integer values from 0 to 100.

The summation index facilitates the systematic and organized listing of terms, ensuring we only include the intended components of the sequence.

Powers of x

Powers of a number, like \(x\), involve multiplying it by itself a specific number of times. In our exercise, this manifests in each term of the form \(x^n\).

• For \(n = 0\), \(x^0\) equals 1, defining the starting point of the sequence.
• As \(n\) increases, \(x\) is raised to higher powers, creating terms like \(x^1, x^2, \ldots, x^{100}\).

Each power of \(x\) represents an exponentially increasing value depending on the magnitude of \(x\), which is key to understanding changes within the sequence. These powers structure the expression, highlighting the exponential pattern inherent in this type of sum.

Greek letter Sigma

The Greek letter Sigma (\(\Sigma\)) is integral to sigma notation, signifying the summation of a sequence of terms. It acts as a compact form of expression for arithmetic series like our example.

• The Sigma symbol accompanies the bounds of the summation, with indexes displayed below and above it. Here, \(\sum_{n=0}^{100}\) defines the range of \(n\).
• Following the Sigma, the general form of the terms to be summed is shown. In our exercise, \(x^n\) represents each term in the summation sequence.

Utilizing the Greek letter Sigma allows for a clearer, more concise presentation of series, simplifying complex arithmetic into a comprehensible notation.