Problem 58 - Algebra and Trigonometry

Question

Find \(x\) so that \(2 x, 3 x+2,\) and \(5 x+3\) are consecutive terms of an arithmetic sequence.

Step-by-Step Solution

Answer

x = 1

Step 1 - Understand Arithmetic Sequence Criteria

In an arithmetic sequence, the difference between consecutive terms is constant. This means that the difference between the second term and the first term should be equal to the difference between the third term and the second term.

Step 2 - Express Differences

Write the differences between the consecutive terms: The difference between the second term \(3x+2\) and the first term \(2x\) is: \[ (3x+2) - 2x = x + 2 \]The difference between the third term \(5x+3\) and the second term \(3x+2\) is:\[ (5x+3) - (3x+2) = 2x + 1 \]

Step 3 - Set Differences Equal

Since the differences must be the same, set the two differences equal to each other:\[ x + 2 = 2x + 1 \]

Step 4 - Solve for x

Isolate \( x \) on one side of the equation:Subtract \( x \) from both sides: \[ x + 2 - x = 2x + 1 - x \]Simplifies to: \[ 2 = x + 1 \]Subtract 1 from both sides: \[ 2 - 1 = x + 1 - 1 \]\[ x = 1 \]

Key Concepts

consecutive terms

In an arithmetic sequence, 'consecutive terms' are adjacent members of the sequence, meaning one follows directly after the other. For example, in the sequence 2, 4, 6, 8, each of these numbers are consecutive terms. The concept of consecutive terms is essential because it helps identify the structure and pattern within the sequence. When given the problem to find a missing value in arithmetic sequences, understanding the relationship between consecutive terms is crucial. It allows us to set up equations based on the differences between these terms.

difference between terms

The 'difference between terms' in an arithmetic sequence remains constant throughout the sequence. This is called the common difference. In the given problem, we are examining the sequence formed by the terms 2x, 3x+2, and 5x+3. To figure out the common difference:

• First, we calculate the difference between the second term (3x+2) and the first term (2x): (3x+2) - 2x = x+2
• Next, we calculate the difference between the third term (5x+3) and the second term (3x+2): (5x+3) - (3x+2) = 2x+1

By setting these differences equal (x+2 = 2x+1), we find the consistency needed for the arithmetic sequence. This constant difference allows us to solve for the variable in question.

solving linear equations

Solving linear equations is a fundamental skill crucial to finding solutions in algebra-based problems. In our problem, after setting the differences equal to each other (x+2 = 2x+1), we need to isolate the variable x. Here are the steps:

• First, subtract x from both sides: x + 2 - x = 2x + 1 - x
• This simplifies to: 2 = x + 1
• Next, subtract 1 from both sides to isolate x: 2 - 1 = x + 1 - 1
• This results in: x = 1

By following these straightforward steps—subtracting and simplifying—you can solve for x. This method is applicable to many algebraic problems, making it a valuable technique in students' mathematical toolkits.