Problem 74 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Verify that for any three consecutive integers, the sum of the smallest and largest is equal to twice the middle integer. [Hint: Use \(n, n+1\), and \(n+2\) to represent the three consecutive integers.]

Step-by-Step Solution

Answer

Sum of smallest and largest = 2 * middle integer; verified by algebra.

Step 1: Define Consecutive Integers

Let the three consecutive integers be represented as: \(n, n+1, n+2\). Here, \(n\) is the smallest integer, \(n+1\) is the middle integer, and \(n+2\) is the largest integer.

Step 2: Find the Sum of the Smallest and Largest

The sum of the smallest integer \(n\) and the largest integer \(n+2\) is given by: \(n + (n+2) = 2n + 2\).

Step 3: Calculate Twice the Middle Integer

The middle integer is \(n+1\), so twice this integer is calculated as: \(2(n+1) = 2n + 2\).

Step 4: Compare Both Sides

Both expressions from Step 2 and Step 3 are identical, i.e., \(2n + 2 = 2n + 2\). This confirms that the sum of the smallest and largest integer is indeed equal to twice the middle integer.

Key Concepts

Algebraic Approach

To tackle a problem using an algebraic approach, we leverage algebraic expressions to represent unknown quantities. In the case of consecutive integers, this method simplifies the problem by expressing each integer as a term related to a common variable. For example:

• The smallest integer is denoted as \( n \).
• The next consecutive integer is \( n + 1 \).
• The largest consecutive integer is \( n + 2 \).

This representation allows us to manipulate these terms algebraically, making it straightforward to explore relationships between them. The algebraic approach is powerful because it turns a problem initially described in words into solvable equations. It also reveals patterns and relationships, making it easier to generalize results.

Integer Verification

Integer verification involves checking if a proposed relationship holds true for integers. In our exercise, we need to verify that the sum of the smallest and largest of three consecutive numbers equals twice the middle one. By applying algebra, we find:

• Sum of smallest and largest: \( n + (n + 2) = 2n + 2 \)
• Twice the middle integer: \( 2(n + 1) = 2n + 2 \)

Both expressions simplify to \( 2n + 2 \). This matching outcome verifies the relationship's validity. Integer verification ensures that algebraic solutions adhere to logical expectations, confirming correctness and consistency across different values. This consistency is crucial in mathematics, teaching students how patterns and formulas apply universally across numbers.

Sum of Integers

When exploring the sum of integers, we focus on understanding how different operations affect the overall sum. This exercise sheds light on how the choice of consecutive integers impacts their sums and relationships. For three consecutive integers \( n, n + 1, n + 2 \), important observations include:

• The sum of the smallest and largest integers is \( 2n + 2 \).
• Doubling the middle integer \( n + 1 \) results in \( 2n + 2 \) too.

Recognizing these sums reinforce the understanding of symmetry and balance within consecutive numbers. **Sum of Integers** concepts go beyond calculations by fostering a deeper comprehension of how integers interact in linear sequences. This understanding can be useful in various mathematical scenarios involving sequences and series.