Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{4} k^{2} $$

Step-by-Step Solution

Answer

The sum of the series is 30.

Step 1: Understand the Series

The series given is \( \sum_{k=1}^{4} k^2 \). It denotes the sum of squares of numbers from 1 to 4. This is an arithmetic series with the formula indicating the terms squared and summed.

Step 2: List the Terms

List all terms in the series as per the given limits of summation. For \( k = 1 \), the term is \( 1^2 \); for \( k = 2 \), the term is \( 2^2 \); for \( k = 3 \), the term is \( 3^2 \); and for \( k = 4 \), the term is \( 4^2 \). This results in the terms: \( 1, 4, 9, \) and \( 16 \).

Step 3: Write the Series as a Sum

Now write the entire series in expanded form: \( 1^2 + 2^2 + 3^2 + 4^2 \) which translates to \( 1 + 4 + 9 + 16 \).

Step 4: Calculate the Sum

Add the series' terms: \( 1 + 4 + 9 + 16 \). Compute stepwise: \( 1 + 4 = 5 \), \( 5 + 9 = 14 \), and finally, \( 14 + 16 = 30 \). Thus, the sum of the series is 30.

Key Concepts

Understanding Summation Notation

Summation notation is a method used to express the addition of a sequence of numbers in a concise form. It is represented by the Greek letter sigma (\( \Sigma \)). This notation includes an expression to be summed, an index of summation, and the limits of summation. For instance, in the series \( \sum_{k=1}^{4} k^2 \):

• The summation symbol \( \Sigma \) indicates that you are summing a series of numbers.
• \( k \) is the index of summation, starting at 1 and ending at 4.
• \( k^2 \) tells you what to do with each value of \( k \) (square it, in this case).

Each number from 1 to 4 is plugged into the expression \( k^2 \) and then all the results are added together. Summation notation is incredibly useful for describing series concisely, especially when dealing with large sets of numbers.

The Process of Series Expansion

The concept of series expansion involves rewriting a series expressed in summation notation as the full set of terms. This is the opposite of using summation notation and gives you a tangible list to actually work with some numbers. Let's examine this process with the series \( \sum_{k=1}^{4} k^2 \):

• For \( k = 1 \), the term is \( 1^2 = 1 \).
• For \( k = 2 \), the term is \( 2^2 = 4 \).
• For \( k = 3 \), the term is \( 3^2 = 9 \).
• For \( k = 4 \), the term is \( 4^2 = 16 \).

By expanding the series, it becomes \( 1 + 4 + 9 + 16 \). This step is crucial as it lays down the numbers you will actually sum up to find the series' total.

Calculating the Sum of Squares

The sum of squares refers to the sum of each number in a sequence squared. This calculation is often involved in statistics and probability, as well as in arithmetic series problems like the one at hand. The task is straightforward once you have expanded the series from the summation notation.In the problem \( \sum_{k=1}^{4} k^2 \), once expanded, the sequence is given as \( 1 + 4 + 9 + 16 \). To compute the sum:

• Start by adding the first two squares: \( 1 + 4 = 5 \).
• Take the result and add the next number: \( 5 + 9 = 14 \).
• Finally, add the last square: \( 14 + 16 = 30 \).

This gives you the total sum of the squares, which is 30 in this case. Understanding this process is key to applying it to more complex sequences and problems.