Problem 48 - Precalculus: Mathematics for Calculus

Question

Find the sum. $$\sum_{k=1}^{4} k^{2}$$

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understand the summation

The notation \( \sum_{k=1}^{4} k^{2} \) represents the sum of the squares of the integers from 1 to 4. This means we need to calculate the square of each integer within this range and then sum those squares together.

Step 2: Calculate each square

Calculate \( k^2 \) for each integer value of \( k \) from 1 to 4: - \( 1^2 = 1 \) - \( 2^2 = 4 \) - \( 3^2 = 9 \) - \( 4^2 = 16 \)

Step 3: Sum the squares

Add the squares calculated in the previous step: \( 1 + 4 + 9 + 16 = 30 \)

Key Concepts

Sum of Squares

In mathematics, the "sum of squares" refers to the summation of each number in a set, where each number is raised to the power of two. Imagine you are tasked with finding the sum of squares of the integers from 1 to 4. What you have to do is simple: you first square each number in the range, then add all the resulting squares together. This process is symbolized by the phrase "sum of squares."

In our example, the sum of squares for the integers 1 through 4 can be denoted as:

• Calculate each square:
  • \(1^2 = 1\)
  • \(2^2 = 4\)
  • \(3^2 = 9\)
  • \(4^2 = 16\)
• Sum them all together:
  • \(1 + 4 + 9 + 16 = 30\)

This concise summation gives a final total of 30, which is the sum of squares of integers from 1 to 4.

Integer Sequence

An integer sequence is nothing more complex than a list of integers arranged in a specific order. In mathematics, sequences are often used to simplify and manage complex calculations by working with ordered elements step-by-step.

In our exercise, we deal with the sequence \(1, 2, 3, 4\). This set is called a sequence because it follows a specific, straightforward order: each number is precisely one greater than the previous. Too often, narrows a broad problem into a more manageable one. Once numbers are aligned in this way, they can be modified, such as squared individually and summed, as required by our task. This kind of structured approach is especially valuable when dealing with more extensive sequences in advanced mathematics.

Mathematical Notation

Mathematical notation is a language used to write down mathematical ideas and concepts formally. It's like a universal language for math that helps mathematicians communicate clearly and precisely without lengthy explanations. You might think of it like a "short-hand" used by mathematicians everywhere.

Consider our example \( \sum_{k=1}^{4} k^{2} \). This notation breaks down into several parts:

• "\( \sum \)" indicates a summation, meaning we are adding something up.
• "\( k = 1 \)" to "\( k = 4 \)" shows the range of summation, dictating that this sum runs from 1 through 4.
• "\( k^2 \)" tells us what we're summing: the square of \( k \).

This concise expression allows one to state a potentially verbose sum in a neat and efficient manner. Understanding how to read and use these symbols is critical in solving most mathematical problems, simplifying both the expression and the solution process.