Nth Term: Understanding the Pattern of 0, 2, 6, 12
The sequence 0, 2, 6, 12, ... is a classic example of an arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term. In this article, we will explore the concept of nth term and how it relates to this sequence.
What is the Nth Term?
The nth term of a sequence is a formula that allows us to calculate any term in the sequence using its position or index (n). In other words, it's a way to express each term in the sequence as a function of its position.
The Pattern of 0, 2, 6, 12
Let's take a closer look at the sequence 0, 2, 6, 12, ... . We can notice that each term is increasing by 2, 4, 6, 8, ... , which is an increment of 2 more than the previous increment.
Finding the Nth Term
To find the nth term of this sequence, we can use the following formula:
an = a1 + (n - 1) * d
where an is the nth term, a1 is the first term (0), n is the position of the term, and d is the common difference (2).
Plugging in the values, we get:
an = 0 + (n - 1) * 2 an = 2n - 2
This is the nth term formula for the sequence 0, 2, 6, 12, ... .
Examples
Let's use the nth term formula to calculate some terms in the sequence:
- Find the 5th term: an = 2(5) - 2 = 8
- Find the 10th term: an = 2(10) - 2 = 18
Conclusion
The sequence 0, 2, 6, 12, ... is a simple yet interesting example of an arithmetic sequence. By finding the nth term, we can easily calculate any term in the sequence using its position. This concept has numerous applications in mathematics, physics, engineering, and other fields, making it an essential tool for problem-solving and analysis.
