Understanding the nth Term of an Arithmetic Sequence
In mathematics, an arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. One of the most important concepts in arithmetic sequences is the nth term, which refers to the formula that allows us to find any term in the sequence.
The Formula for the nth Term
The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1)d
Where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
Example 1: Finding the nth Term
Let's consider the sequence: 2, 6, 12, 20, ...
To find the nth term, we need to identify the first term (a1) and the common difference (d).
a1 = 2 d = 6 - 2 = 4
Now, we can use the formula to find the nth term:
an = 2 + (n - 1)4
an = 2 + 4n - 4
an = 4n - 2
Finding Specific Terms
Using the formula, we can find any term in the sequence. For example, let's find the 10th term:
an = 4(10) - 2 an = 40 - 2 an = 38
Therefore, the 10th term in the sequence is 38.
Conclusion
In conclusion, the nth term formula is a powerful tool for finding any term in an arithmetic sequence. By understanding this formula, we can easily calculate any term in the sequence, making it a valuable skill in mathematics.
Practice Exercises
- Find the 15th term in the sequence: 3, 7, 11, 15, ...
- Find the nth term formula for the sequence: 1, 4, 9, 16, ...
Solutions
- an = 3 + (15 - 1)4 = 3 + 56 = 59
- an = 1 + (n - 1)3 = 1 + 3n - 3 = 3n - 2
