Problem 39 - Algebra and Trigonometry

Question

Find the sum. $$\sum_{k=1}^{4} k$$

Step-by-Step Solution

Answer

The sum is 10.

Step 1: Understanding the Summation

The problem requires finding the sum of the series represented by \( \sum_{k=1}^{4} k \). This notation means we add up the numbers from 1 to 4.

Step 2: Listing the Terms

Identify the actual numbers you need to sum up by replacing \(k\) with each integer value from the lower to upper bound: 1, 2, 3, and 4.

Step 3: Calculating the Sum

Add the terms together: \(1 + 2 + 3 + 4\). First, sum 1 and 2 to get 3. Then add 3 to get 6. Finally, add 4 to get a total of 10.

Step 4: Verifying the Result

You can verify the sum quickly by using the formula for the sum of the first \(n\) natural numbers, \( \frac{n(n+1)}{2} \). Here, \(n = 4\): \( \frac{4(4+1)}{2} = 10 \). This matches our calculated sum.

Key Concepts

Understanding Series

In mathematics, a series is the sum of terms of a sequence of numbers. When we talk about a series, we are looking at the entire sum created by this list of numbers rather than examining them individually.
For example, the series from our original exercise was expressed with the notation \( \sum_{k=1}^{4} k \).
This is a summation notation indicating that all terms from 1 to 4 should be added together. Series can greatly simplify the process of adding long sequences by using a compact representation.

Instead of writing out \(1 + 2 + 3 + 4\), the series expression provides an easy-to-read structure. This becomes even more useful with longer lists. Series can also be extended to infinite sums, providing insights into calculus and other advanced mathematical studies.

• Finite series have a specific number of terms.
• Infinite series continue indefinitely.

Understanding series and summation notation is essential for developing skills in algebra and beyond.

Natural Numbers

Natural numbers are the set of positive integers that start from 1 and continue infinitely.
They do not include fractions, decimals, or negative numbers.
In our exercise, we worked with natural numbers 1 through 4 (i.e., \(1, 2, 3, 4\)).

Natural numbers are the simplest form of numbers used for counting and ordering. They are the basis for more complex numbers and are crucial in many areas of mathematics, such as arithmetic and number theory.

• The sequence starts at 1, often seen as the 'first' natural number.
• There is no upper limit, so the sequence goes on indefinitely.

Natural numbers are integral in defining other mathematical concepts, such as whole numbers, integers, and rational numbers.

Summation Formula

The summation formula is a powerful tool that helps in calculating the sum of a sequence without individually adding each number.
For instance, when finding the sum of the first \(n\) natural numbers, like in our example, we use the formula \( \frac{n(n+1)}{2} \).

This formula provides a quick and efficient way to find the sum without having to manually add each term. Let's see how it works:

• For \(n = 4\), you plug into the formula: \( \frac{4(4+1)}{2} = 10 \).
• The same result is achieved as when you summed manually, confirming the formula's accuracy.

The summation formula is beneficial when dealing with larger sequences since it reduces time and errors, making calculations straightforward. Understanding how to apply this formula offers a strong foundation for further mathematical exploration involving sequences and series.