The formula for finding a sequence depends on the type of sequence you are working with. There are several common types of sequences, each with its own formula:
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: a<sub>n</sub> = a<sub>1</sub> + (n - 1)d
Where:
- a<sub>n</sub> is the nth term of the sequence
- a<sub>1</sub> is the first term of the sequence
- d is the common difference
- n is the position of the term in the sequence
Example:
Consider the sequence: 2, 5, 8, 11, 14...
- a<sub>1</sub> = 2
- d = 3 (5 - 2 = 3, 8 - 5 = 3, etc.)
To find the 10th term (a<sub>10</sub>):
- a<sub>10</sub> = 2 + (10 - 1)3 = 2 + 27 = 29
Geometric Sequences
A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio.
- Formula: a<sub>n</sub> = a<sub>1</sub> r<sup>(n - 1)</sup>*
Where:
- a<sub>n</sub> is the nth term of the sequence
- a<sub>1</sub> is the first term of the sequence
- r is the common ratio
- n is the position of the term in the sequence
Example:
Consider the sequence: 3, 6, 12, 24, 48...
- a<sub>1</sub> = 3
- r = 2 (6 / 3 = 2, 12 / 6 = 2, etc.)
To find the 7th term (a<sub>7</sub>):
- a<sub>7</sub> = 3 2<sup>(7 - 1)</sup> = 3 2<sup>6</sup> = 3 * 64 = 192
Fibonacci Sequence
The Fibonacci sequence is a sequence where each term is the sum of the two preceding terms.
- Formula: F<sub>n</sub> = F<sub>n-1</sub> + F<sub>n-2</sub>
Where:
- F<sub>n</sub> is the nth term of the sequence
- F<sub>n-1</sub> is the (n-1)th term of the sequence
- F<sub>n-2</sub> is the (n-2)th term of the sequence
Example:
The Fibonacci sequence starts with 0 and 1:
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
To find the 10th term (F<sub>10</sub>), you would add the 9th and 8th terms:
- F<sub>10</sub> = F<sub>9</sub> + F<sub>8</sub> = 34 + 21 = 55
Other Types of Sequences
There are many other types of sequences, including:
- Harmonic Sequences: The reciprocals of the terms in an arithmetic sequence.
- Geometric-Arithmetic Sequences: A combination of arithmetic and geometric sequences.
- Recursively Defined Sequences: Sequences defined by a formula that relates each term to previous terms.
For these sequences, there may be specific formulas or methods to find the nth term.
Note: It is important to identify the type of sequence you are working with before applying any formula.
