Solving an Arithmetic Progression Problem: Finding x and the First Eight Terms

Arithmetic progressions (APs) are sequences of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This article will walk you through a problem involving an arithmetic progression, demonstrating how to find the value of (x) and the first eight terms of the progression.

Given the first three terms of an arithmetic progression are (8 - x), (3x), and (4x 1), we need to find the value of (x) and the first eight terms of the progression.

Step 1: Introduce the Common Difference (d)

The common difference, (d), in an arithmetic progression is the difference between consecutive terms. For this problem:

[d (3x) - (8 - x) (4x 1) - (3x)]

Let's simplify these expressions to find (d):

[d 3x - 8 x 4x 1 - 3x]

[d 4x - 8 x 1]

Step 2: Solve for x

Now, we set the two expressions for (d) equal to each other:

[4x - 8 x 1]

To solve for (x), we need to subtract (x) from both sides and add 8 to both sides:

[3x 9]

Divide both sides by 3:

[x 3]

Step 3: Find the First Term

The first term of the arithmetic progression is given by (8 - x). Substituting the value of (x 3):

[8 - 3 5]

Step 4: Determine the Common Difference

The common difference (d) can be found by substituting (x 3) into the expression (4x - 8):

[4(3) - 8 12 - 8 4]

Step 5: Find the First Eight Terms

Using the first term and the common difference, we can find the first eight terms of the arithmetic progression:

[5, 5 4, 5 2(4), 5 3(4), 5 4(4), 5 5(4), 5 6(4), 5 7(4)]

Simplifying each term, we get:

5, 9, 13, 17, 21, 25, 29, 33

Conclusion

By solving the given arithmetic progression problem, we have found (x 3), the common difference (d 4), and the first eight terms of the progression, which are 5, 9, 13, 17, 21, 25, 29, and 33.