Understanding and Generating Arithmetic Sequences

The content provided discusses the generation of a sequence of numbers in which each term is obtained by adding a constant difference to the previous term. This type of sequence is known as an arithmetic progression (AP).

In the given sequence, 0, 3, 6, 9, 12, 15, 18, 21, the first term is 0 and the common difference (d) is 3. This pattern can be easily recognized and continued, leading to the next term being 24, as demonstrated below.

Identifying the Pattern in the Sequence

The initial sequence is as follows:

We notice each term increases by 3. To predict the next term in the sequence:

Following this pattern, subsequent terms would be 27, 30, etc.

Generating the Arithmetic Sequence

Given the sequence 6, 9, 12, 15, 18, 21, we can determine the common difference (d).

The common difference (d) is:

Given the first term (a) is 6 and the common difference (d) is 3, the nth term of the arithmetic progression can be calculated using the formula:

[ t_n a (n-1)d ]

Using this formula for the 9th term (n9), we get:

[ t_9 6 (9-1) times 3 6 24 30 ]

However, it is important to note that this specific sequence is given to end at 21, and the next term would be 24, as previously mentioned.

General Form of an Arithmetic Sequence

The general form of an arithmetic sequence where the first term is a and the common difference is d, the nth term is given by:

[ t_n a (n-1)d ]

For the sequence 3n, where n starts from 2 (to avoid the first term 0), the sequence becomes 6, 9, 12, 15, 18, 21. The nth term of this sequence is:

[ t_n 3n - 3 ]

Or written as a formula, it is:

[ t_n 3n - 3 ]

Conclusion

Understanding the generation of an arithmetic sequence is crucial for identifying and predicting the next terms in a sequence. The common difference, the first term, and the formula for the nth term are key elements to consider. Applying these principles, we can easily generate and understand various arithmetic sequences.