Solving Arithmetic Progressions: Finding First Terms and Common Differences

Arithmetic progressions (AP) are sequences in which the difference between consecutive terms is constant. This common difference is a defining feature of arithmetic sequences. In this article, we will explore how to find the first term and the common difference of an AP, using given terms of the sequence.

Understanding the Problem

Consider an arithmetic progression with the first term (a) and common difference (d). The (n)th term (t_n) of the AP can be expressed using the formula:

[t_n a (n - 1)d]

Solving for the First Term and Common Difference

Given:

(t_4 15) (t_{10} 27)

Let's find the common difference (d) first.

Step 1: Find the Common Difference (d)

We have:

[t_{10} - t_4 (a 9d) - (a 3d) 27 - 15]

This simplifies to:

[6d 12]

Solving for (d):

[d 2]

Step 2: Find the First Term (a)

Using the value of (d), we can now find the first term (a) using the formula for the 4th term:

[t_4 a 3d 15]

Solving for (a):

[a 3(2) 15]

[a 6 15]

[a 15 - 6]

[a 9]

Thus, the first term (a) is 9, and the common difference (d) is 2.

Step 3: Find the 17th Term (t_{17})

To find the 17th term, we use the formula for the (n)th term of an AP:

[t_{17} a 16d]

Substituting the values of (a) and (d):

[t_{17} 9 16(2) 9 32 41]

General Rule of an Arithmetic Sequence

The general term rule of the arithmetic sequence can be written as:

[t_n 9 (n - 1)2]

Or simplified further:

[t_n 2n 7]

Additional Example: Finding Common Difference and First Term

Let's also solve a similar problem where the 4th and 10th terms are given:

Given

(t_4 25) (t_{10} 13)

Using the same method:

Step 1: Find the Common Difference (d)

[t_{10} - t_4 (a 9d) - (a 3d) 13 - 25]

[6d -12]

[d -2]

Step 2: Find the First Term (a)

Using the 4th term formula:

[25 a 3(-2)]

[25 a - 6]

[a 31]

Step 3: Find the 17th Term (t_{17})

[t_{17} a 16d 31 16(-2) 31 - 32 -1]

The arithmetic sequence is: 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, -1 …

The general term rule can be written as:

[t_n 31 - 2(n - 1)]

Or simplified further:

[t_n 33 - 2n]

Conclusion

By understanding the structure of an arithmetic progression and applying the appropriate formulas, it is straightforward to determine the first term and the common difference, as well as any term in the sequence. This article has provided clear examples and steps to solve for these values.