Problem 41 - Algebra and Trigonometry

Question

\(41-48\) Find the sum. $$ \sum_{k=1}^{4} k $$

Step-by-Step Solution

Answer

The sum is 10.

Step 1: Understand the Sigma Notation

The symbol \(\sum_{k=1}^{4}\) denotes a summation, meaning you add up all the terms when \(k\) changes from 1 to 4. The expression after the summation sign, \(k\), indicates the terms to be added.

Step 2: Substitute Values of k

Now, substitute the values \(k = 1, 2, 3,\) and \(4\) into the expression \(k\). This gives the series of terms to add: \(1, 2, 3, 4\).

Step 3: Perform Addition

Add the terms obtained from substituting different \(k\) values: \(1 + 2 + 3 + 4\).

Step 4: Calculate the Sum

Evaluate the addition: \(1 + 2 = 3\), then \(3 + 3 = 6\), and finally, \(6 + 4 = 10\).

Key Concepts

Summation

Summation is a mathematical concept represented by the sigma notation \(\sum\). It involves adding a series of numbers according to a specified rule or formula.
The sigma (\(\sum\)) symbol is used to denote this kind of structured addition. When you see \(\sum_{k=1}^{4}\), it signifies that you should add all terms from \(k = 1\) to \(k = 4\).
This gives you the sequence of numbers that you need to work with. Summation is commonly used in mathematics to simplify the process of calculating the total of several terms without writing out each one individually.

• The lower number beneath the sigma (here it's 1) is the starting point, telling us the first value of \(k\).
• The number on top of the sigma (here it's 4) informs us the final value of \(k\).
• The expression next to the sigma shows what exactly to add. In our case, it is simply \(k\).

By understanding summation, you can quickly determine what numbers to add and where to start and finish in a sequence.

Series Addition

Series addition is the actual process of adding the numbers in the sequence specified by the summation notation. After determining which numbers are to be added, list them out.
For the exercise in question, we substitute numbers from 1 to 4 into our sequence. This produces the list of terms: 1, 2, 3, 4.
Once the terms are identified, the next step is to add them together.

• The series in this case starts with the number 1, progresses to 2, and continues to 3, and finally to 4.
• Add these terms sequentially to ensure accuracy and completeness.
• Writing them out helps visualize the progression from one term to the next.

Series addition methodically combines these terms, helping you confirm each step is correctly performed before moving to the next.

Evaluate Sum

Evaluating the sum refers to the final stage where you actually calculate the total of the series you have assembled from the previous steps.
After listing out the terms of the series as 1, 2, 3, and 4 from the sigma notation, you move on to the addition.
Start with the first number, add it to the second, continue with the next, and so on, ensuring not to miss any term.

• Begin by adding the first two terms: \(1 + 2 = 3\).
• Add the result to the next number in the series: \(3 + 3 = 6\).
• Finally, combine this result with the last term: \(6 + 4 = 10\).

By following this sequential approach, evaluating the sum becomes a straightforward task that helps consolidate your understanding of working through a series.