Problem 40 - Precalculus: Mathematical 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 Notation

The notation \( \sum_{k=1}^{4} k^2 \) stands for the sum of the squares of the first four positive integers. This means we want to calculate \( 1^2 + 2^2 + 3^2 + 4^2 \).

Step 2: Calculate Individual Squares

First, compute the square of each integer from 1 to 4. \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), and \( 4^2 = 16 \).

Step 3: Add the Squares

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

Step 4: Find the Sum

Perform the addition: \(1 + 4 = 5\), \(5 + 9 = 14\), and finally, \(14 + 16 = 30\).

Key Concepts

Series notation

Series notation is a compact and efficient way of representing the sum of a sequence. If you've ever come across the symbol \( \sum \), you're looking at sigma notation. This represents the idea of summing multiple terms. The expression typically has an index variable, like \( k \), which changes value in each term of the sum. In our example, \( \sum_{k=1}^{4} k^{2} \) means we're summing the squares of integers from 1 to 4.
The subscript \( k=1 \) indicates that \( k \) starts at 1, while the superscript 4 means \( k \) goes up to 4. Thus, \( k \) takes each integer value from 1 to 4, and those integers are then squared and summed together.
This series notation is particularly useful when dealing with large sums, as it provides a neat and straightforward way to represent them without writing out each individual term.

Arithmetic operations

Arithmetic operations are the fundamental processes of mathematics: addition, subtraction, multiplication, and division. In the context of our exercise, performing arithmetic operations helps us find the sum of the squared numbers.Firstly, we square each number, which is a form of multiplication (a number times itself). For example, getting \( 2^2 = 4 \).
After finding the squares of each number from 1 to 4, we then perform addition, which is the step-by-step process of adding those results together: \( 1 + 4 + 9 + 16 \). These operations are essential to solving many mathematical problems, as they allow us to transform and combine numbers in meaningful ways.
Here are some key points about arithmetic operations:

• Addition is combining two or more numbers to get a total.
• Each number in addition is called an addend and the result is the sum.
• In multiplication, the result is called the product.

Understanding these operations is crucial as they form the basis for solving more complex mathematical problems.

Exponents

Exponents are a way of representing repeated multiplication of the same number. This concept appears in our exercise as we consider \( k^2 \), which means multiplying \( k \) by itself (\( k \times k \)).Exponents have their own rules that make calculations easier. For instance, \( a^m \times a^n = a^{m+n} \) combines two powers of the same base by adding their exponents.
In our calculation, you multiply:

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

Using exponents makes expressions more manageable and compact, which is especially useful for larger numbers.
Mastery of exponents is essential, as they are commonly used in algebra, calculus, and other mathematical fields.