Problem 31 - College Algebra

Question

Find the indicated sum. $$ \sum_{n=1}^{6} n(n-2) $$

Step-by-Step Solution

Answer

The sum is 49.

Step 1: Understand the Problem

We need to find the sum of the expression \( n(n-2) \) for values of \( n \) ranging from 1 to 6. This involves evaluating the expression for each integer value from 1 through 6 and then summing the results.

Step 2: Expand the Expression

The given expression \( n(n-2) \) can be expanded to \( n^2 - 2n \). This rewriting may help simplify the calculations.

Step 3: Evaluate the Expression for Each Integer

Substitute each integer from 1 to 6 into the expression \( n(n-2) \) and calculate the value:- For \( n = 1 \), \( 1(1-2) = 1 \times -1 = -1 \)- For \( n = 2 \), \( 2(2-2) = 2 \times 0 = 0 \)- For \( n = 3 \), \( 3(3-2) = 3 \times 1 = 3 \)- For \( n = 4 \), \( 4(4-2) = 4 \times 2 = 8 \)- For \( n = 5 \), \( 5(5-2) = 5 \times 3 = 15 \)- For \( n = 6 \), \( 6(6-2) = 6 \times 4 = 24 \)

Step 4: Calculate the Sum

Add up all the calculated values from Step 3:\[ -1 + 0 + 3 + 8 + 15 + 24 = 49 \]

Step 5: Conclusion

The sum of \( n(n-2) \) as \( n \) goes from 1 to 6 is 49. Ensure that all values are correctly added and verify the calculations if necessary.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical statements that contain numbers, variables, and operators. They represent a value or a collection of values. In the context of the problem, the expression \( n(n-2) \) is a specific type of algebraic expression that involves the variable \( n \). This expression represents a sequence of calculations that depend on the value of \( n \). When dealing with algebraic expressions, it's important to understand the role of each component - numbers (like 2), variables (like \( n \)), and operators (like multiplication and subtraction). By mastering these components, you can efficiently manipulate and evaluate expressions, which is a key skill in algebra.

Expansion

Expansion in mathematics refers to rewriting an expression in an alternative form, which often simplifies the calculation or provides clearer insight. In our current problem, expanding \( n(n-2) \) to \( n^2 - 2n \) helps us break down the computation into more familiar terms involving powers and linear expressions.

• The term \( n^2 \) denotes a square of the integer, which is easily recognizable.
• The term \(-2n \) involves simple multiplication and subtraction.

By expanding expressions, you can more easily apply known mathematical principles and streamline the process of finding a solution. Learn to spot opportunities for expanding different types of algebraic expressions, as it is a valuable tool in problem solving.

Integer Evaluation

Integer evaluation is the process of substituting integers into an expression and calculating the result. This method is crucial when determining specific values for problems involving sequences or series.In this exercise, we evaluated the expression \( n(n-2) \) for integer values from 1 to 6. Here's how it works:

• Replace \( n \) with a specific integer.
• Perform the necessary arithmetic operations to compute the expression's value.

For instance, when \( n = 3 \), we substitute 3 into the expression to get \( 3 imes (3-2) = 3 \times 1 = 3 \). Repeat this for each integer up to 6. Understanding integer evaluation ensures you can handle calculations with precision and speed.

Mathematical Problem Solving

Mathematical problem solving is about applying a structured, logical approach to find solutions to math problems. This exercise provides a good example of problem-solving using summation.Here's how you tackle such problems:

• **Identify the problem:** Understand the expression and the range of values for \( n \).
• **Break it down:** Use techniques like expansion to simplify the problem.
• **Compute:** Evaluate the expression for each relevant integer.
• **Summarize the findings:** Add up all the evaluated expressions to find the sum.

By following these steps, you develop critical thinking skills and become better at tackling various mathematical challenges. It helps create clear and manageable steps that lead to the solution, making even complex problems easier to solve.