Problem 42 - Algebra and Trigonometry

Question

\(41-48\) Find the sum. $$ \sum_{k=1}^{4} k^{2} $$

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understanding the Sigma Notation

The given expression is the sigma notation for the sum of squares of the first four natural numbers. The expression is \( \sum_{k=1}^{4} k^2 \) which indicates we need to calculate the square of each number from 1 to 4 and then sum them up.

Step 2: Calculating Individual Squares

Calculate the squares of all numbers from 1 to 4:- \(1^2 = 1\)- \(2^2 = 4\)- \(3^2 = 9\)- \(4^2 = 16\)

Step 3: Summing the Squares

Now add all the calculated squares together:\[1 + 4 + 9 + 16 = 30\]

Step 4: Final Sum Result

The sum of the squares \( \sum_{k=1}^{4} k^2 \) is 30.

Key Concepts

Sigma Notation

Sigma notation is a concise and powerful way to express the sum of a series of terms. It uses the Greek letter \( \Sigma \), representing "summation," to denote the operation of adding a sequence of numbers together. In the expression \( \sum_{k=1}^{4} k^2 \):

• The \( k=1 \) at the bottom of the Sigma implies the starting value of the index \( k \).
• The \( 4 \) at the top signifies the last value that \( k \) will take on.
• The \( k^2 \) after the Sigma specifies the term that is to be summed, here it is the square of \( k \).

This notation simplifies writing the sum of a finite series, especially useful for large series. By reading \( \sum_{k=1}^{4} k^2 \), we know we need to calculate \( 1^2 + 2^2 + 3^2 + 4^2 \) and find their sum. This saves us from writing each term individually, making mathematical writing and problem solving easier.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and increasing indefinitely: 1, 2, 3, 4, etc. These are the most basic numbers we use every day to count objects.
Natural numbers form the foundation for more complex number systems and are important in defining series or sequences, like the one in our exercise.
In \( \sum_{k=1}^{4} k^2 \), the numbers 1, 2, 3, and 4 are natural numbers. They indicate the terms we need to square and then sum.
Because they follow a regular pattern, natural numbers are often used in sequences, making it easier to perform operations like summation using sigma notation.

Square of a Number

The square of a number is the result of multiplying that number by itself. It is an important concept in mathematics used in various calculations and real-world applications.
For example, if you have a number \( n \), its square is represented as \( n^2 \), calculated by \( n \times n \).
In our exercise, we find the squares of 1, 2, 3, and 4. Here's a quick breakdown:

• \(1^2 = 1 \cdot 1 = 1\)
• \(2^2 = 2 \cdot 2 = 4\)
• \(3^2 = 3 \cdot 3 = 9\)
• \(4^2 = 4 \cdot 4 = 16\)

These squares are then summed to solve our sigma notation problem. Squaring numbers is a fundamental arithmetic operation used frequently in geometry, physics, and computer science for various calculations.