Solving an Arithmetic Progression to Find the Common Difference

In this article, we will explore an interesting problem in arithmetic progression where the first term is given as 8, and the tenth term is double the second term. Through a series of algebraic steps, we will learn how to find the common difference and determine the terms of the progression.

Solving the Problem: Finding the Common Difference

The problem we are considering is as follows: If the first term of an arithmetic progression (AP) is 8 (denoted as T1 8) and the tenth term is double the second term, what is the common difference, d? Let's solve it step-by-step.

Given Information

The first term, T1 8. The tenth term, T10, is double the second term, T2.

Step-by-Step Solution

Let's start by expressing the terms of the AP in terms of the first term, T1 8, and the common difference, d.

Second Term (T2)

The second term, T2, can be written as:

T2 T1 d 8 d

Tenth Term (T10)

The tenth term, T10, is given by:

T10 T1 9d 8 9d

Given Condition

According to the problem, the tenth term is double the second term:

T10 2T2

Substituting the values, we get:

8 9d 2(8 d)

Expanding and simplifying:

8 9d 16 2d

Subtracting 2d from both sides:

7d 8

Dividing both sides by 7:

d 8/7

Conclusion

The common difference d is 8/7. Using this, we can find the specific terms of the AP:

Second Term (T2)

T2 8 d

8 8/7 64/7

Tenth Term (T10)

T10 8 9d

8 9(8/7) 8 72/7 160/7

Thus, the AP can be written as:

8, 64/7, 72/7, 80/7, 88/7, 96/7, 104/7, 160/7, ...

General Term Rule for the Sequence

The general term of the sequence can be written as:

Tn 8 (n-1)(8/7)

8 8n/7 - 8/7

(8n 56 - 8)/7

(8n 48)/7

Conclusion

In summary, we have solved the arithmetic progression problem where the first term is 8, and the tenth term is double the second term. The common difference d was found to be 8/7. The sequence was then derived, and the general term rule was established.