The rule for the nth term of an arithmetic sequence is an = a1 + (n - 1)d, where:
- an represents the value of the nth term.
- a1 represents the first term of the sequence.
- n represents the position of the term in the sequence.
- d represents the common difference between consecutive terms.
To find the nth term of a sequence, follow these steps:
- Calculate the common difference (d): Subtract any two consecutive terms in the sequence.
- Multiply the term number (n) by the common difference (d): This gives you the difference between the nth term and the first term.
- Add the result from step 2 to the first term (a1): This gives you the value of the nth term (an).
Example:
Consider the arithmetic sequence: 2, 5, 8, 11, ...
- a1 = 2 (the first term)
- d = 5 - 2 = 3 (the common difference)
To find the 5th term (a5):
- *(n - 1)d = (5 - 1) 3 = 12**
- a1 + (n - 1)d = 2 + 12 = 14
Therefore, the 5th term of the sequence is 14.
Key takeaways:
- The nth term formula is a powerful tool for finding any term in an arithmetic sequence.
- It simplifies the process of determining terms in a sequence by providing a direct relationship between the term number and its value.
- Understanding the formula allows for predicting future terms in a sequence and analyzing its properties.
