Determining the Next Term in an Arithmetic Progression with Variables

Mathematical series are a fascinating aspect of algebra, and understanding their properties can provide valuable insights into more complex mathematical problems. One common type of series in mathematics is an arithmetic progression (AP), where each term after the first is obtained by adding a constant, called the common difference, to the preceding term. In this article, we will explore how to find the next term in an arithmetic progression when the terms are given in terms of variables.

Understanding Arithmetic Progression (AP)

Recall the general formula for an arithmetic progression:

T1, T2, T3, ...: a, a d, a 2d, ...

Where:

a is the first term of the arithmetic series. d is the common difference between each of the terms.

Given the terms:

T1 6x - 2 T2 4x 12 T3 11x 3

and knowing that these terms form an arithmetic progression, we need to find the next term, T4.

Establishing the Condition for an AP

The middle term of an arithmetic progression is the average of the other two terms. Therefore:

2T2 T1 T3

Substituting the given terms:

2(4x 12) (6x - 2) (11x 3)

Simplifying the equation:

8x 24 17x 1

Reorganizing the terms to solve for x:

24 - 1 17x - 8x

23 9x

x 23/9

Substituting the Value of x

Now, we can substitute the value of x back into the expressions for T1, T2, and T3:

T1 6(23/9) - 2 (138/9) - (18/9) 120/9 40/3

T2 4(23/9) 12 (92/9) - (108/9) 200/9

T3 11(23/9) 3 (253/9) - (27/9) 280/9

Finding the Common Difference

The common difference d can be calculated as:

d T2 - T1 p>Substituting the values:

d (200/9) - (40/3) (200/9) - (120/9) 80/9

Calculating the Next Term

The next term, T4, is found by adding the common difference d to T3:

T4 T3 d (280/9) (80/9) 360/9 40

Therefore, the next term in the series is:

boxed{40}