4 Consecutive Integers That Add Up To 254

Finding Four Consecutive Integers

This article explores the problem of finding four consecutive integers that sum to 254. We will approach this problem using algebraic methods, verifying our solution and then generalizing the approach to solve similar problems involving different numbers of consecutive integers and sums.

Understanding the Problem

Consecutive integers are whole numbers that follow each other in order, without any gaps. For example, 1, 2, 3, and 4 are consecutive integers. To represent four consecutive integers algebraically, we can let ‘x’ represent the first integer. The next three consecutive integers would then be represented as x + 1, x + 2, and x + 3. The equation representing the sum of these four consecutive integers equaling 254 is: x + (x + 1) + (x + 2) + (x + 3) = 254.

Solving the Equation

To solve the equation x + (x + 1) + (x + 2) + (x + 3) = 254, we can simplify and solve for x. Several methods can be employed. We will demonstrate the solution using simplification:

1. Combine like terms: 4x + 6 = 254
2. Subtract 6 from both sides: 4x = 248
3. Divide both sides by 4: x = 62

Therefore, the first integer (x) is 62. The four consecutive integers are 62, 63, 64, and 65.

Verification and Interpretation of Results

Let’s verify the solution by substituting the integers back into the original equation: 62 + 63 + 64 + 65 = 254. The equation holds true, confirming our solution. The solution means that the four consecutive integers 62, 63, 64, and 65 add up to 254. There are no alternative solutions since consecutive integers are uniquely defined.

Generalizing the Problem

To find the sum of ‘n’ consecutive integers, starting with ‘x’, the general equation is: x + (x+1) + (x+2) + … + (x + n-1) = S, where S is the sum. This can be simplified to nx + n(n-1)/2 = S. This formula can be used to solve for different sums and numbers of consecutive integers by substituting the known values and solving for the unknown variable.

This problem is related to the sum of an arithmetic series, where the common difference is 1. The formula for the sum of an arithmetic series is S = n/2[2a + (n-1)d], where ‘n’ is the number of terms, ‘a’ is the first term, and ‘d’ is the common difference. In our case, d=1, making it directly applicable to consecutive integers.

Visual Representation

The following table displays the four consecutive integers and their running total:

A flowchart illustrating the solution process:

• Define the problem: Find four consecutive integers that sum to 254.
• Represent algebraically: x + (x+1) + (x+2) + (x+3) = 254
• Simplify the equation: 4x + 6 = 254
• Solve for x: x = 62
• Determine the integers: 62, 63, 64, 65
• Verify the solution: 62 + 63 + 64 + 65 = 254

A bar chart illustrating the integers: Imagine a bar chart with four bars. The first bar represents 62, the second 63, the third 64, and the fourth 65. Each bar’s height corresponds to its numerical value, clearly showing the consecutive nature and relative magnitudes of the integers.

Exploring Related Concepts

Consecutive integers form an arithmetic progression with a common difference of 1. The average of these consecutive integers is the middle value (or the average of the two middle values if there’s an even number of integers). In this case, the average is (62+63+64+65)/4 = 63.5. The sum of consecutive integers is directly related to their average; the sum is simply the average multiplied by the number of integers.