Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated sum. \(\sum_{n=1}^{6} n(n-2)\)

Step-by-Step Solution

Answer

The sum is 49.

Step 1: Understand the Summation

We need to find the sum of the expression \(n(n-2)\) from \(n=1\) to \(n=6\). This means we will calculate the value of the expression for each integer from 1 to 6 and then add them together.

Step 2: Calculate Each Term

Calculate \(n(n-2)\) for each \(n\) from 1 to 6:- For \(n=1\): \(1(1-2) = 1(-1) = -1\)- For \(n=2\): \(2(2-2) = 2(0) = 0\)- For \(n=3\): \(3(3-2) = 3(1) = 3\)- For \(n=4\): \(4(4-2) = 4(2) = 8\)- For \(n=5\): \(5(5-2) = 5(3) = 15\)- For \(n=6\): \(6(6-2) = 6(4) = 24\)

Step 3: Sum the Terms

Add the results from Step 2:\(-1 + 0 + 3 + 8 + 15 + 24 = 49\)Thus, the sum of the series is 49.

Key Concepts

Series and Sequences

In algebra, a series is the summation of the terms of a sequence, which is an ordered list of numbers. Understanding series and sequences is crucial when dealing with problems that involve adding up numbers following a specific pattern. The expression \( \sum_{n=1}^{6} n(n-2) \) denotes a series where each term is calculated by the formula \( n(n-2) \) from \( n = 1 \) to \( n = 6 \) and then added up together.

A sequence dictates the order and structure, while a series is concerned with the total sum of this order. Here, our sequence begins at 1 and progresses incrementally by 1 until it reaches 6. This straightforward arithmetic sequence forms the backbone of our calculation. The sequence \( \lbrace 1, 2, 3, 4, 5, 6 \rbrace \) when mapped through the formula \( n(n-2) \) gives a processed sequence of terms, the sum of which becomes the targeted outcome in this exercise.

Recognizing how each element in a series is derived from a sequence helps decode the pattern and find solutions efficiently.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics, allowing for the representation of numbers and variables through operations such as addition, subtraction, multiplication, and division. In this exercise, the expression \( n(n-2) \) is key. Here, \( n \) represents an integer term from the sequence, while \( n-2 \) adjusts that integer to construct a new value.

In the expression \( n(n-2) \):

• \( n \) is the variable that changes with each term of the sequence.
• \( n-2 \) makes the expression quadratic and transforms the input to yield a sequence with interesting properties.

By substituting each integer from the sequence into this expression, you'll calculate and notice the unique relationship each result has with its input. Learning to manipulate and simplify these expressions is critical for recognizing patterns and solving complex algebra problems.

Step-by-Step Problem Solving

Breaking down problems into manageable steps is a vital strategy for solving mathematical exercises efficiently. In our case, solving \( \sum_{n=1}^{6} n(n-2) \) was handled through a clear and sequential approach.

**Step 1: Understanding the Problem**
- Recognize that you need to find the total sum of an expression evaluated over a series of terms.
- Identify the range and the formula to apply.

**Step 2: Calculating Individual Terms**
- Evaluate \( n(n-2) \) for each \( n \) within the defined range to obtain specific results:
- \( n=1 \): \( 1(-1) = -1 \)
- \( n=2 \): \( 2(0) = 0 \)
- \( n=3 \): \( 3(1) = 3 \)
- \( n=4 \): \( 4(2) = 8 \)
- \( n=5 \): \( 5(3) = 15 \)
- \( n=6 \): \( 6(4) = 24 \)

**Step 3: Summing the Results**
- Adding up your calculated terms to find the overall sum: \( -1 + 0 + 3 + 8 + 15 + 24 = 49 \).

Following these steps ensures each part of the problem is understood and tackled methodically, making the solution easier to verify and comprehend.