A succession, more commonly known as a sequence in mathematics, is an ordered list of numbers that follow a specific pattern or rule. There are different types of sequences, but the two most common are arithmetic sequences and geometric sequences.
Arithmetic Sequences
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted as $d$
General Formula
The general formula for the $n$-th term of an arithmetic sequence is:
$a_n = a_1 + (n-1)d$
Where:
- $a_n$ is the $n$-th term
- $a_1$ is the first term
- $n$ is the term number
- $d$ is the common difference
Example
Consider the arithmetic sequence: 2, 5, 8, 11, 14, …
Here, $a_1 = 2$ and $d = 3$. To find the 5th term ($a_5$), we use the formula:
$a_5 = 2 + (5-1) times 3 = 2 + 12 = 14$
Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as $r$
General Formula
The general formula for the $n$-th term of a geometric sequence is:
$a_n = a_1 times r^{(n-1)}$
Where:
- $a_n$ is the $n$-th term
- $a_1$ is the first term
- $n$ is the term number
- $r$ is the common ratio
Example
Consider the geometric sequence: 3, 6, 12, 24, 48, …
Here, $a_1 = 3$ and $r = 2$. To find the 4th term ($a_4$), we use the formula:
$a_4 = 3 times 2^{(4-1)} = 3 times 2^3 = 3 times 8 = 24$
Conclusion
Understanding the formulas for arithmetic and geometric sequences helps in identifying patterns and solving problems related to sequences in mathematics. These concepts are widely used in various fields, including finance, computer science, and physics, making them essential tools for mathematical analysis.