Solving for x in an Arithmetic Progression Using Given Terms

Arithmetic progressions, or APs, are sequences of numbers in which the differences between consecutive terms are constant. This article will guide you through the process of solving for x and determining the common difference in an AP using the given terms: x3, 2x - 3, and x5.

Understanding the Problem

The problem states that x3, 2x - 3, and x5 are consecutive terms of an arithmetic progression. We need to find the value of x and the common difference.

Step 1: Setting Up the Equation

Given that the terms are consecutive in an arithmetic progression, the difference between each pair of consecutive terms is constant. Therefore, we can set up the equation:

x5 - (2x - 3) (2x - 3) - x3

Step 2: Simplifying the Equation

Substitute the terms x3, 2x - 3, and x5 into the equation:

x5 - 2x 3 2x - 3 - x3

Now, rearrange the terms to group like terms:

x5 x3 - 2x - 2x 3 3 0

Simplify the equation:

x5 x3 - 4x 6 0

Step 3: Solving for x

Given that the terms are consecutive in an arithmetic progression, we can also directly use the relationship between the terms to solve for x:

2x - 3 - x3 x5 - 2x - 3

Rearrange the equation:

x5 x3 - 4x 6 0

Step 4: Simplifying Further

By inspection or solving the equation, we find:

2x - 3 - x 3

2x - x 3 3

x 7

Now, substituting x 7 back into the original terms:

x3 73 10

2x - 3 2(7) - 3 11

x5 75 12

Hence, the terms are 10, 11, and 12.

Step 5: Identifying the Common Difference

The common difference, d, is the difference between each pair of consecutive terms:

d 11 - 10 1

Hence, the common difference is 1.

Conclusion

The value of x that satisfies the given terms in an arithmetic progression is 7, and the common difference is 1.