Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=1}^{6} n^{3} $$

Step-by-Step Solution

Answer

The sum is 441.

Step 1: Understand the given expression

We are working with the expression \( \sum_{n=1}^{6} n^3 \). This means that we need to find the sum of cubes of natural numbers from 1 to 6.

Step 2: Expand the Series

Write the series expanded as follows: \(1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3\). This shows each term in the sequence with 6 terms as indicated by \( n \) ranging from 1 to 6.

Step 3: Calculate Each Term

Calculate each term individually: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\), \(5^3 = 125\), \(6^3 = 216\).

Step 4: Sum the Terms

Add all the calculated values together: \(1 + 8 + 27 + 64 + 125 + 216 = 441\). This is the sum of the series.

Key Concepts

Sum of Cubes

When we talk about the sum of cubes, we are looking into the idea of adding together a series of numbers that have each been raised to the third power. This is known as cubing a number. For example, the cube of 2 is written as \(2^3\) and calculated as \(2 \times 2 \times 2 = 8\). The sum of cubes from 1 to a certain number \(n\) follows a particular formula that looks like this: \[ \left( \frac{n(n+1)}{2} \right)^2\]This formula can save time by avoiding the need to manually calculate each cube and add them together. However, understanding how to expand and calculate each cube separately provides a solid grasp of the arithmetic process involved, especially for learners new to this topic.
To get a better understanding, let's visualize it with numbers. For the series from 1 to 6, we get:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• \(6^3 = 216\)

By adding these together, the sum of cubes is found to be 441. This approach helps solidify the concept through practice.

Series Expansion

Series expansion involves writing out each term in a series all the way through to the last term. This is a crucial step when working with arithmetic series as it allows us to see and understand each part of the sequence involved. Consider the series given by \( \sum_{n=1}^{6} n^3 \). Expanding the series means expressing it as an individual sequence of terms. In this case, the series \( \sum_{n=1}^{6} n^3 \) becomes:

• \(1^3\)
• \(2^3\)
• \(3^3\)
• \(4^3\)
• \(5^3\)
• \(6^3\)

This step allows us to see each arithmetic operation that needs to be performed, building a clear path to solving the problem. It gives clarity by showing the incremental progression and helps avoid errors that might occur if steps are skipped.
Through series expansion, not only do we ensure correctness, but we also gain a deeper insight into the entire mathematical operation. It's an excellent way to visualize and understand how the final sum is achieved when all parts of the series come together.

Calculating Series Sum

Once the series has been expanded by listing each term, the next step is to calculate the series sum, which involves adding these terms together. This procedure gives us the final result. The concept of calculating a series sum effectively comes down to performing the arithmetic calculation step by step. For the arithmetic series where each term is a cube from 1 to 6, it's essential to tackle it in stages:1. **Calculate each term individually:** - \(1^3 = 1\) - \(2^3 = 8\) - \(3^3 = 27\) - \(4^3 = 64\) - \(5^3 = 125\) - \(6^3 = 216\)2. **Add the results of each calculation:** - Start from the first term, adding each subsequent term: - \(1 + 8 + 27 + 64 + 125 + 216 = 441\)
This approach ensures accuracy by isolating each step, allowing for verification and an understanding of how each part of the series contributes to the total sum. By following this method, you build a comprehensive understanding of how to tackle arithmetic series, ensuring you can approach similar problems with confidence.