Problem 31 - Algebra for College Students

Question

$$ \sum_{i=1}^{5} i^{3} \quad 225 $$

Step-by-Step Solution

Answer

The sum \(\sum_{i=1}^{5} i^3\) equals 225.

Step 1: Understanding the Summation Notation

The expression \( \sum_{i=1}^{5} i^3 \) represents the sum of the cubes of the first 5 positive integers. This means we need to calculate \( 1^3 + 2^3 + 3^3 + 4^3 + 5^3 \).

Step 2: Calculating Each Cube

Compute the cubes of each individual integer from 1 to 5.- \(1^3 = 1\)- \(2^3 = 8\)- \(3^3 = 27\)- \(4^3 = 64\)- \(5^3 = 125\)This provides us the individual values to sum up.

Step 3: Summing the Cubes

Add up all the cubes calculated in Step 2: \[ 1 + 8 + 27 + 64 + 125 \]

Step 4: Calculating the Sum

Perform the addition: - \(1 + 8 = 9\)- \(9 + 27 = 36\)- \(36 + 64 = 100\)- \(100 + 125 = 225\)Thus, the sum of the cubes is 225.

Step 5: Verifying the Solution

Check that the calculated sum \(225\) matches the given result in the problem statement, confirming the correctness of our calculations.

Key Concepts

Cubic Numbers

Cubic numbers are the result of multiplying an integer by itself twice. In mathematical terms, the cube of a number \( i \) is represented as \( i^3 \). This is not only a fundamental mathematical operation but also incredibly important in understanding various mathematical concepts. When we hear about cubic numbers, we visualize perfect cubes that correspond to the multiplication of numbers in three dimensions.

For instance, if we take \( 2 \), its cube is \( 2^3 = 2 \times 2 \times 2 = 8 \). Each result represents the volume of a cube with sides of length \( 2 \). To quickly compute a cubic number:

• Identify the integer you wish to cube.
• Multiply the integer by itself to get the square.
• Multiply the square by the original integer to get the cube.

Understanding cubes provides a platform for solving more complex problems, as in our original exercise of summating multiple cubed numbers.

Arithmetic Series

An arithmetic series involves the summation of a sequence of numbers where the difference between consecutive terms is consistent. In the context of our exercise, while the numbers being cubed do not individually form an arithmetic series, their bases \( (1, 2, 3, 4, 5) \) do have a constant difference of 1.

The structure of an arithmetic series is simple, and it can be very helpful when calculating sums. For example:

• The numbers \( 1, 2, 3, 4, 5 \) form an arithmetic sequence with a common difference of \( 1 \).
• Add the numbers together individually or apply an arithmetic series formula to find the sum.

This understanding can simplify calculations especially when extending the concept to other series, such as summing squares, cubes, or other powers of natural numbers.

Mathematical Calculation

Mathematical calculations often involve breaking down complex problems into more manageable steps. This approach ensures accuracy and prevents errors. When dealing with the summation of cubed numbers, as in the exercise, these types of calculations can be especially useful.The method we followed combined basic arithmetic operations performed step-by-step:

• First, calculate each cubic number individually, ensuring each multiplication sequence (\( 1^3, 2^3, \) etc.) is correct.
• Then, sum the results from the cubes for an aggregate total.

Accuracy in calculations is fundamental as slight errors can lead to incorrect results. This structured approach allows tackling even more complex problems where larger numerical operations are involved. By verifying calculations, as seen in Step 5, the solution's integrity is confirmed, giving confidence in the derived results.