Problem 67 - Algebra for College Students

Question

\sum_{i=1}^{5} i^{2} 55

Step-by-Step Solution

Answer

The sum is 55.

Step 1: Understand the Summation Notation

The given exercise involves a summation notation \( \sum_{i=1}^{5} i^{2} \), which instructs you to sum the squares of the numbers starting from 1 up to 5. Thus, you need to calculate \( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 \).

Step 2: Calculate Each Square

Calculate the square of each individual integer from 1 to 5. - \( 1^2 = 1 \)- \( 2^2 = 4 \)- \( 3^2 = 9 \)- \( 4^2 = 16 \)- \( 5^2 = 25 \)

Step 3: Sum the Squares

Now that you have the squares, add them together:\( 1 + 4 + 9 + 16 + 25 \).

Step 4: Perform the Addition

Add the numbers calculated from the squares: \( 1 + 4 + 9 + 16 + 25 = 55 \).

Step 5: Conclusion

The sum of the squares of the first five positive integers is 55, which completes the exercise.

Key Concepts

Square of a Number

When we talk about squaring a number, we mean multiplying the number by itself. So, if you have a number like 2, squaring it means:

• You multiply 2 by 2

This gives you:

• 4

It's as if you're saying the number appears in a two-dimensional space. Squaring numbers is a basic operation that frequently pops up in math. For example, squaring the number 3 would involve:

• 3 multiplied by 3, resulting in 9.

Squares have some unique properties, such as always being non-negative (since the product of two negative numbers also turns out positive). Understanding this will help you easily follow summation and series exercises.

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. In the case of an exercise appearing in summation notation, we often encounter series where we add things consecutively.
The numbers we are working with follow a predictable pattern. For instance, when examining the summation of squares like in the exercise:

• 1 + 4 + 9 + 16 + 25

Each element in this sequence stems from squaring natural numbers starting from 1.
The arithmetic series helps in summing sequences quickly and efficiently. Series can be both finite (as it stops after a certain number of terms) or infinite. In exercises like ours, the series does have an endpoint, and employing summation notation simplifies representing this sort of structured calculation.

Step-by-step Solutions

Step-by-step solutions provide a roadmap for solving a problem, detailing each action. For a summation and squaring exercise, breaking it down clearly becomes crucial.
Here's how it typically flows:

• Interpret the notation: Understanding what the symbol or expression asks you to find, often starting with smaller, manageable calculations.
• Calculate components: It's best to handle one aspect at a time, like determining individual squares before trying to tackle a sum.
• Combine components: This means putting those individual results together, leading to a final, comprehensive sum.
• Verify your calculations: Double-checking ensures that any errors are caught, giving you confidence in your conclusion.

Each step builds toward a thorough understanding, ensuring not only a correct answer but also a grasp of underlying concepts. This technique is useful across mathematical exercises and real-life problem-solving scenarios.