Understanding the nth Term of the Sequence 3, 6, 9, 12...

In this article, we will explore the rule for the nth term of the sequence 3, 6, 9, 12, .... This sequence is a classic example of an arithmetic sequence, where each term is obtained by adding a constant value, known as the common difference, to the previous term.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. In our sequence, the first term is 3, and the common difference is 3. For an arithmetic sequence, the nth term can be found using the formula:

an a1 (n - 1) d

Applying the Formula to the Given Sequence

Given the first term ((a_1 3)) and the common difference ((d 3)), let's apply the formula:

an 3 (n - 1) 3

Simplifying this equation, we get:

an 3 3n - 3 3n

Therefore, the nth term rule for this sequence is:

an 3n

Alternative Methods to Derive the nth Term

Method 1: Direct Multiplication

Another method to derive the nth term of the sequence is by direct multiplication. As each term is a multiple of 3 and increases by a factor of 3, we can express the nth term as:

nth term n 3

For example:

tThe 1st term: (1 3 3) tThe 2nd term: (2 3 6) tThe 3rd term: (3 3 9) tThe 4th term: (4 3 12)

Method 2: General Formula Using Difference

We can also derive the nth term by using the general formula based on the common difference:

Nth term First term (n - 1) Difference

Substituting the known values:

3n 3 (n - 1) 3

Again, simplifying this will lead us to the same result:

3n 3 3n - 3 3n

Conclusion

In summary, the nth term of the sequence 3, 6, 9, 12, ..., can be expressed by the formula (a_n 3n). This is a straightforward example of an arithmetic sequence, and understanding such sequences is crucial for many areas of mathematics and practical applications. Whether you use the general formula or recognize patterns in the sequence, the result remains consistent.