Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understand the Problem

We need to find the sum of the squares of the first 4 positive integers, meaning we will evaluate the expression \( k^2 \) for \( k = 1 \) to \( k = 4 \).

Step 2: Write Out the Terms

Express each term in the sum: 1. When \( k = 1 \), \( k^2 = 1^2 = 1 \).2. When \( k = 2 \), \( k^2 = 2^2 = 4 \).3. When \( k = 3 \), \( k^2 = 3^2 = 9 \).4. When \( k = 4 \), \( k^2 = 4^2 = 16 \).

Step 3: Calculate the Sum of the Terms

Add up all the terms we wrote out:\[1 + 4 + 9 + 16 = 30\]

Step 4: Conclude the Result

The sum of the squares of the first 4 positive integers is 30. The mathematical representation is:\[\sum_{k=1}^{4} k^{2} = 30\] This concludes our calculation.

Key Concepts

Positive Integers

Positive integers are numbers starting from 1 and increasing to infinity, without including zero or any negative numbers. They are part of a larger group called integers, which also includes zero and negative numbers.
In mathematics, positive integers are often used in counting and ordering. They play a crucial role in various concepts such as sequences, series, and more advanced mathematical theories.

• Properties of Positive Integers: Positive integers are whole numbers greater than zero.
• Simple Examples: 1, 2, 3, 4, 5, 6...
• Usage: Widely used in arithmetic operations, solving equations, and statistical data analysis.

Understanding positive integers is the first step in evaluating sums, especially when working with series or sequences in exercises like summing the squares of numbers.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators (like +, −, ×, ÷) that represents a value. These expressions can be as simple as a single number or as complex as a formula with multiple variables.
In our exercise, the expression is the sum of the squares of integers from 1 to 4: \( \sum_{k=1}^{4} k^{2} \).
Writing mathematical expressions enables us to communicate a mathematical idea clearly and precisely. They help in evaluating calculations systematically.

• Basic Components: Numbers, variables, operators.
• Example: In \( k^2 \), \( k \) is the variable representing the integer values from 1 to 4.
• Benefits: Expressions simplify complex calculations and allow easy manipulation of values to reach a result.

Grasping the construction and purpose of mathematical expressions is vital for solving problems, as these expressions become the building blocks of mathematical reasoning.

Evaluation of Sums

Evaluating sums involves calculating the total of a series of numbers or expressions. This often requires understanding how to sum individual components methodically.
In our example, we evaluate the expression \( \sum_{k=1}^{4} k^{2} \) by calculating each term's square and then adding them together.
Here’s how you generally approach evaluating a sum of squares, step by step:

• Identify the Terms: Determine the values of \( k \) to be used in the expression. For example, from 1 to 4.
• Calculate Each Term: Square each value of \( k \) to get its contribution to the sum.
• Add the Results: Sum the squares of each term to find the final result.

Evaluating sums is a straightforward yet powerful mathematical technique. It transforms a series of calculations into a single, clear answer. Understanding this process allows for more advanced application in algebra and calculus.