Problem 59 - Precalculus Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{k=3}^{100} x^{k}$$

Step-by-Step Solution

Answer

The sum is written as: \( x^3 + x^4 + x^5 + \ldots + x^{100} \).

Step 1: Understand the sigma notation

The given sum, \( \sum_{k=3}^{100} x^k \), indicates that we want to add up terms starting with \( x^3 \) and ending with \( x^{100} \). Each term in this sum is of the form \( x^k \) where \( k \) starts at 3 and goes up to 100.

Step 2: Write out the sequence

To express the sum without sigma notation, we need to list each term from the beginning to the end. This means starting with \( x^3 \), then \( x^4 \), and continuing until \( x^{100} \).

Step 3: Express the entire sum as a sequence

Writing the sum without sigma notation involves explicitly listing all the terms added together: \[ x^3 + x^4 + x^5 + ext{...} + x^{100} \]Here, the terms increase in exponent from 3 to 100.

Key Concepts

Sequence

In mathematics, a sequence is an ordered list of numbers. When dealing with sequences, each term has a specific position and follows a particular pattern. Understanding sequences is essential when working with computations that involve multiple steps.

In the context of the given problem, the sequence arises from terms presented in sigma notation. Sigma notation is a concise way to represent a series, which is the sum of a sequence. Here, the sequence is \(x^3, x^4, x^5, \ldots, x^{100}\), where each term is related by incrementing the exponent by one. This forms an arithmetic progression.

Recognizing the structure within a sequence helps simplify expressions and evaluates sums more effectively. By breaking down the sequence visually or in list form, you can better understand how each part contributes to the whole. This lays the groundwork for the subsequent summation process.

Exponents

Exponents denote repeated multiplication of a base number. For instance, \(x^3\) means multiplying the base \(x\) by itself three times: \(x \times x \times x\). This shorthand is invaluable in mathematics for simplifying expressions and calculations.

In the provided problem, each term in the sequence includes an exponent of the variable \(x\). The exponents start from 3 and go up to 100. This incremental pattern is crucial as it dictates the progression of the sequence. By gradually increasing the exponent, the problem reflects how repetitive multiplication can expand numbers rapidly.

Understanding how exponents function enables you to manipulate mathematical expressions more easily and see patterns that may arise in problem-solving scenarios involving power sums like the one presented here.

Summation

Summation refers to the addition of a sequence of numbers. The sigma notation is a compact form used to specify the sum of a sequence without listing all the terms individually. In mathematics, transforming from sigma notation to an explicit list of terms requires expressing the whole sequence.

In our scenario, we're dealing with the summation of a sequence using the sigma notation \(\sum_{k=3}^{100} x^k \). To write it out without sigma notation, each term from the sequence must be added cumulatively, starting from \(x^3\) and ending at \(x^{100}\). This creates a long addition expression like \(x^3 + x^4 + \ldots + x^{100}\).

Understanding summation helps in evaluating expressions and breaking them into manageable parts. It highlights the importance of process accuracy when generating terms in large expressions, allowing complex operations to reduce down to fundamental arithmetic steps.