Discovering the Pattern of Sequences: Unveiling the Mystery behind 0 12 24 36 48 60 and Beyond

The sequence 0 12 24 36 48 60... is a fascinating example of an arithmetic sequence, a fundamental concept in mathematics. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant (common difference) to the previous term. In this case, the common difference is 12.

Understanding the Sequence

The pattern in the sequence is as follows:

0 12 12 12 12 24 24 12 36 36 12 48 48 12 60 60 12 72

To find the next number in the sequence, we simply add 12 to the previous number.

The Explicit Formula for Arithmetic Sequences

The explicit formula for an arithmetic sequence is given by:

an a1 (n - 1) cdot; d

where:

an is the nth term in the sequence, a1 is the first term in the sequence, d is the common difference.

In our case, with the first term a1 0 and the common difference d 12, we can calculate the 7th term as:

a7 0 (7 - 1) cdot; 12 0 6 cdot; 12 72

Conclusion of the Sequence

The sequence: 0, 12, 24, 36, 48, 60... continues as:

60 12 72 72 12 84 84 12 96 96 12 108 108 12 120

And so on.

Frequently Asked Questions

What is the next number in the sequence after 0 12 24 36 48 60?

The next number in the sequence is 72, as each term is 12 more than the previous term.

How do you generate the sequence 0 12 24 36 48 60...

Each term is generated by adding 12 to the previous term. Starting from 0, the terms are 0, 12, 24, 36, 48, and 60, and the next term is 72.

Can you provide the formula to find the nth term of the sequence?

The formula for the nth term of the sequence is an (n - 1) * 12 (with the first term being a1 0).