How to Find the Difference of an Arithmetic Progression

In this article, we will explore how to find the common difference of an arithmetic progression (AP) using given terms. The terms A6 and A15 of an AP are given as A6 4 / (2 - √3) and A15 -2√3. We will step through the process of calculating the common difference, d, using these terms.

Given Information

The terms A6 and A15 are given by:

A6 4 / (2 - √3) A15 -2√3

Steps to Find the Common Difference, d

We know that in an AP, the nth term can be found using the formula:

An An-1 d

Thus, we can express A6 and A15 in terms of A5 and A14, respectively:

A6 A5 d

A15 A14 d

Combining these, we get:

Step 1: Express A6 and A15 in Terms of d

We use the given terms to express A5 and A14

A6 A5 d 4 / (2 - √3) ……….1

A15 A14 d -2√3 …………2

Step 2: Subtract Equation 1 from Equation 2

A15 - A6 A14 - A5

Substituting the given values:

-2√3 - 4 / (2 - √3) 9d

Step 3: Simplify the Equation

Let's simplify the left-hand side of the equation:

9d -2√3 - 4 / (2 - √3)

First, rationalize the denominator:

4 / (2 - √3) * (2 √3) / (2 √3) 4(2 √3) / (4 - 3) 4(2 √3)

So, the equation becomes:

9d -2√3 - 4(2 √3) -2√3 - 8 - 4√3

9d -2√3 - 8 - 4√3 -10√3 - 8

Rearrange to solve for d:

9d -2√3 - 8

d (-2√3 - 8) / 9 2/9 - 8/9√3

Step 4: Final Expression for the Common Difference

The common difference, d, is:

d 2/9 - 8√3/9 2 - 8√3/9

Thus, the common difference of the arithmetic progression is d 2 - 8√3/9.

Practice for the Reader

If you try to solve the problem yourself and get stuck, let us know. We are here to help!

Conclusion

By following the steps outlined in this article, you can calculate the common difference of an arithmetic progression given specific terms. Remember to rationalize denominators and simplify expressions carefully. The common difference, d, is a fundamental concept in understanding the properties of sequences and series.