Problem 42 - 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 notation

The expression \( \sum_{k=1}^{4} k^2 \) means we need to find the sum of the squares of the integers from 1 to 4. This involves calculating \( 1^2 + 2^2 + 3^2 + 4^2 \).

Step 2: Calculate each individual square

First, we calculate each of the squares: - \( 1^2 = 1 \) - \( 2^2 = 4 \) - \( 3^2 = 9 \) - \( 4^2 = 16 \)

Step 3: Sum the individual squares

Add the squares calculated from Step 2 together: \( 1 + 4 + 9 + 16 \).

Step 4: Calculate the total sum

Perform the addition: \( 1 + 4 = 5 \) \( 5 + 9 = 14 \) \( 14 + 16 = 30 \). Thus, the total sum is 30.

Key Concepts

Summing Squares

Summing squares is all about adding the squares of numbers together. A square of a number is simply that number multiplied by itself. For instance, the square of 2 is calculated as \(2^2 = 4\). When you see the notation \(\sum_{k=1}^{4} k^2\), it tells you to find the sum of the squares of the numbers 1 through 4.

• Start with squaring each number individually: \(1^2, 2^2, 3^2,\) and \(4^2\).
• Then, add up all these squares to get your answer.

In the problem we are working on, 1 squared is 1, 2 squared is 4, 3 squared is 9, and 4 squared is 16. Adding these together, we find that the sum of these squares is 30. This step-by-step approach makes it easier to handle the summation of squares and can be applied to any similar kind of problem.

Integer Sequences

Integer sequences involve a list or series of numbers that follow a specific pattern. Understanding integer sequences is crucial when working with summation problems, especially when dealing with indices and series. In the example given, the sequence is a simple list of the integers from 1 to 4.

• The sequence starts at 1 and increases by 1 each time, ending at 4.
• Each term in this sequence is squared, as indicated by the \(k^2\) in our summation notation.

Recognizing these sequences will help you know exactly which numbers to utilize in your calculations. When you become familiar with various sequences—like arithmetic sequences where each term has a consistent difference—you gain an important tool for evaluating sums and other mathematical expressions.

Basic Arithmetic Operations

Basic arithmetic operations are foundational to solving many types of mathematical problems. These include addition, subtraction, multiplication, and division. In this exercise, you primarily use addition, as you are summing the values after squaring each integer.

• First, **square** each number: multiply the number by itself.
• Then, **add up** these squares to find the sum.
• It's important to perform each operation step-by-step, ensuring you're doing the calculations correctly at each stage.

For instance, after you square the numbers, add them together in stages: 1 plus 4 equals 5, then 5 plus 9 equals 14, and finally add 14 and 16 to get 30. By tackling each part methodically, arithmetic problems become more manageable and less prone to mistakes.