Arithmetic and Geometric Sequences
What is a numerical sequence?
A sequence is an ordered list of numbers defined by a rule. Each term depends on its position and follows a specific pattern. Two of the most important types of sequences are arithmetic and geometric sequences.
Understanding these two patterns is essential because they appear in many areas of mathematics, from basic algebra to more advanced topics.
Arithmetic sequence
In an arithmetic sequence, the difference between consecutive terms is constant. This constant value is called the common difference. $$ a_n - a_{n-1} = d $$
This means that each term is obtained by adding the same number to the previous one $$ a_n = a_{n-1} + d $$
Example
This is an arithmetic sequence
$$ 3 \ , \ 5 \ , \ 7 \ , \ 9 \ , \ 11 \ , \ 13 \ \dots $$
The common difference is
$$ d = 2 $$
So each term is found by adding 2 to the previous one
$$ a_n = a_{n-1} + 2 $$
Starting from the first term a1=3, the sequence develops step by step
$$ a_2 = 3 + 2 = 5 \\ a_3 = 5 + 2 = 7 \\ a_4 = 7 + 2 = 9 \\ a_5 = 9 + 2 = 11 \\ a_6 = 11 + 2 = 13 $$
Geometric sequence
In a geometric sequence, the ratio between consecutive terms is constant. This constant factor is called the common ratio. $$ \frac{a_n}{a_{n-1}} = q $$
This means that each term is obtained by multiplying the previous one by the same number $$ a_n = a_{n-1} \cdot q $$
Example
This is a geometric sequence
$$ 2 \ , \ 4 \ , \ 8 \ , \ 16 \ , \ 32 \ , \ 64 \ \dots $$
The common ratio is
$$ q = 2 $$
So each term is found by multiplying the previous one by 2
$$ a_n = a_{n-1} \cdot 2 $$
Starting from the first term a1=2, the sequence develops step by step
$$ a_2 = 2 \cdot 2 = 4 \\ a_3 = 4 \cdot 2 = 8 \\ a_4 = 8 \cdot 2 = 16 \\ a_5 = 16 \cdot 2 = 32 \\ a_6 = 32 \cdot 2 = 64 $$
And so on
