Problem 65 - College Algebra with Modeling and Visualization

Question

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{7} k^{3}$$

Step-by-Step Solution

Answer

The sum of the series \( \sum_{k=1}^{7} k^3 \) is 784.

Step 1: Understand the Series

The given problem requires evaluating the series \( \sum_{k=1}^{7} k^3 \). This means we need to calculate the sum of cubes of each integer from 1 to 7 inclusive.

Step 2: Write Out the Terms

To evaluate the series, first write down each term: \( 1^3, 2^3, 3^3, 4^3, 5^3, 6^3, \) and \( 7^3 \). These terms originate from cubing each integer in the range from 1 to 7.

Step 3: Compute Individual Cubes

Compute each cube separately: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\), \(5^3 = 125\), \(6^3 = 216\), and \(7^3 = 343\).

Step 4: Sum the Cubes

Add the results from Step 3 together: \(1 + 8 + 27 + 64 + 125 + 216 + 343\).

Step 5: Calculate the Final Result

Perform the addition: \(1 + 8 + 27 + 64 + 125 + 216 + 343 = 784\). Thus, the sum of the series is 784.

Key Concepts

Understanding Summation Notation

Summation notation is a concise way of representing the addition of a sequence of numbers, especially when dealing with series. The Greek letter sigma (\( \sum \)) symbolizes this process. In the context of our exercise, the notation \( \sum_{k=1}^{7} k^3 \) represents the sum of the cubes of integers from 1 to 7. Let's break it down:

• \( k \) is the index of summation, which takes integer values from the starting point (1) to the ending point (7).
• The expression \( k^3 \) inside the summation represents each term in the sequence.
• The entire expression instructs us to sum these terms as \( k \) changes from 1 to 7.

Understanding this notation simplifies tackling series problems and is a fundamental skill in algebra and calculus.

Exploring the Cube of Integers

Cubing a number means multiplying it by itself twice more. For example, \( k^3 \) is \( k \times k \times k \). Here's how it works with some integers:

• \( 1^3 = 1 \times 1 \times 1 = 1 \)
• \( 2^3 = 2 \times 2 \times 2 = 8 \)
• \( 3^3 = 3 \times 3 \times 3 = 27 \)
• Continuing this pattern helps us understand exponential growth, where each step drastically increases the value.

Cubes of integers grow quickly, which helps demonstrate the power of exponential functions through practical examples in mathematical problems.

Introduction to Algebraic Series

An algebraic series is a collection of numbers added together, where each element is defined by an algebraic expression. In this exercise, the series \( \sum_{k=1}^{7} k^3 \) is a classic example. Each term in the series is determined by cubing the integers. This exercise reveals a recurring structure inherent in algebraic expressions.

Algebraic series can also be expressed using formulas for quick calculations, particularly useful for larger series. This efficiency becomes crucial in complex mathematical applications and functions.

Applying a Step-by-Step Solution

The step-by-step approach is vital in solving mathematical series and understanding the solution thoroughly. Here's how it applies to our problem:\( \sum_{k=1}^{7} k^3 \):

• Identify the task: Recognize the need to sum the cubes of numbers from 1 to 7.
• List each term: Write down \(1^3, 2^3, 3^3, 4^3, 5^3, 6^3, 7^3\) to see what we're dealing with.
• Calculate each cube: Perform the individual multiplications: \(1, 8, 27, 64, 125, 216, 343\).
• Add the results: Combine these results for the final sum: \(784\).

Following structured steps ensures clarity and success in tackling difficult problems, preventing errors and building confidence in mathematical processes.