Problem 57 - Algebra and Trigonometry

Question

Find \(x\) so that \(x+3,2 x+1,\) and \(5 x+2\) are consecutive terms of an arithmetic sequence.

Step-by-Step Solution

Answer

The value of x is -1.5.

Step 1 - Understand the Condition for Arithmetic Sequence

In an arithmetic sequence, the difference between consecutive terms is constant. Thus, for the terms to be consecutive in an arithmetic sequence, the difference between the first and second term must be equal to the difference between the second and third terms.

Step 2 - Equate the Differences

We need to equate the differences between the consecutive terms: For the first and second terms: (2x + 1) - (x + 3) = x - 2. For the second and third terms: (5x + 2) - (2x + 1) = 3x + 1. Equating these two equations results in: x - 2 = 3x + 1.

Step 3 - Solve for x

Solving the equation for x: x - 2 = 3x + 1 -2 - 1 = 3x - x -3 = 2x x = -1.5

Step 4 - Verify the Solution

We substitute x back into the terms to verify the solution: The first term: -1.5 + 3 = 1.5, The second term: 2(-1.5) + 1 = -2 + 1 = -1, The third term: 5(-1.5) + 2 = -7.5 + 2 = -5.5 The differences -1.5 and 1 do match: 1.5 - (-1) = -1 - (-5), proving an arithmetic sequence.

Key Concepts

consecutive terms

Consecutive terms in an arithmetic sequence have a very special property. The difference between each term and the next is always the same. For example, in the sequence 2, 5, 8, 11, the difference between each term is 3. Understanding this consistent interval helps identify arithmetic sequences and solve related problems.
To check if terms are consecutive in such a sequence, you can always look at the differences between pairs of terms. In our problem, we have three terms: \(x+3\), \(2x+1\), and \(5x+2\). To confirm they are in an arithmetic sequence, you need to ensure that the difference between the first and second term matches the difference between the second and third term.

solving for x

To solve for x, you need to use the property of consecutive terms in an arithmetic sequence. You start by equating the differences.

• First, calculate the difference between the first and second terms:
(2x + 1) - (x + 3) = x - 2.

• Second, calculate the difference between the second and third terms:
(5x + 2) - (2x + 1) = 3x + 1.

By equating the two differences, you get:
x - 2 = 3x + 1.
Now, solve for x by isolating it on one side:

• -2 - 1 = 3x - x
• -3 = 2x
• x = -1.5

With this value of x, you can determine the actual terms of the sequence.

sequence verification

Verifying the solution is a crucial step in ensuring the correctness of your results. Substituting x back into the terms allows you to check if the calculated differences are consistent:

• First term: \(-1.5 + 3 = 1.5\)
• Second term: \(2(-1.5) + 1 = -2 + 1 = -1\)
• Third term: \(5(-1.5) + 2 = -7.5 + 2 = -5.5\)

Now, compute the differences to check consistency:

• Difference between first and second terms: \(-1 - 1.5 = -2.5\)
• Difference between second and third terms: \(-5.5 - -1 = -4.5\)

By verifying the differences, you confirm that the calculated terms are indeed part of an arithmetic sequence. This process of substituting back and checking the differences ensures that your solution is accurate.