The formula for the nth term depends on the type of pattern. One common type of pattern is an arithmetic sequence.
Arithmetic Sequences
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.
Formula for the nth term of an Arithmetic Sequence
The formula for finding the nth term (an) of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an = the nth term we want to find
- a1 = the first term of the sequence
- n = the position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.)
- d = the common difference between consecutive terms
Example:
Consider the arithmetic sequence: 2, 5, 8, 11,... (as given in the reference 21-Sept-2023, but with a more common notation.)
Here, a1 = 2 and d = 3 (since 5 - 2 = 3, 8 - 5 = 3, and so on).
To find the 5th term (a5):
a5 = 2 + (5 - 1) 3
a5 = 2 + (4) 3
a5 = 2 + 12
a5 = 14
Therefore, the 5th term in the sequence is 14.
Other Types of Patterns
It is important to note that the above formula applies specifically to arithmetic sequences. Other types of patterns (e.g., geometric sequences, quadratic sequences) will have different formulas for determining the nth term. Without knowing the specific type of pattern, a general formula cannot be provided.
