What is the 4th Term of an Arithmetic Progression if the First Term is 3 and the Common Difference is 6?

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This article will guide you through the process of finding the 4th term of an arithmetic progression when the first term and the common difference are given.

Understanding the Arithmetic Progression

Given the first term ((a_1)) is 3 and the common difference ((d)) is 6, we can determine the 4th term of the sequence. Let's break down the process step-by-step.

Step-by-Step Calculation

Step 2: Use the formula for the nth term of an arithmetic progression:

[a_n a_1 (n-1)d]

Step 3: Substitute the values of (a_1), (d), and (n) (where (n 4)) into the formula:

[a_4 3 (4-1)6]

Step 4: Simplify the expression:

[a_4 3 3middot;6]

Step 5: Calculate the product:

[a_4 3 18]

Step 6: Add the numbers together:

[a_4 21]

Summary of the Arithmetic Sequence

The arithmetic sequence with the first term 3 and the common difference 6 is:

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, ...

The general term rule for this arithmetic sequence is:

[a_n 6n - 3]

Conclusion

By using the formula for the nth term of an arithmetic progression and substituting the given values, we have successfully determined that the 4th term of the given arithmetic progression is 21. This process can be applied to any arithmetic sequence to find any term regardless of its position in the sequence.