Arithmetic Sequence
What is an arithmetic sequence
A sequence $a_{n}$ is called an arithmetic sequence (or arithmetic progression) if it consists of numbers arranged in such a way that the difference between any term and the one before it is constant. It is characterized by terms of the form:
\[a_{1} , a_{2} , \ldots , a_{n} \text{with} a_{n} - a_{n - 1} = d\]- By convention, the first term of an arithmetic progression is typically indexed with $n = 1.$
- $d$ represents the difference between two consecutive terms in an arithmetic progression, and it is known as the common difference.
- If $d > 0$, the progression is increasing.
- If $d < 0$, the progression is decreasing.
- If $d = 0$, the progression is constant.
Let’s consider, for example, the sequence of non-negative even numbers:
An arithmetic sequence can also be defined using a recursive formula:
\[a_{n} = a_{1} + n \cdot d \text{where} a_{1} , d \in \mathbb{R}\]
An arithmetic progression exhibits a characteristic stepwise pattern, where the height of each step corresponds to the common difference between consecutive terms in the sequence.
In an arithmetic progression, each term $a_{n}$ is obtained by adding the first term $a_{1}$ to the product of the common difference $d$ and $( n - 1 )$. This gives the general formula for the $n$-th term:
\[a_{n} = a_{1} + ( n - 1 ) \cdot d \text{for} n \geq 1\]This formula allows you to compute any term in the sequence directly, without listing all the previous ones.
Example
Let’s define an arithmetic sequence with first term $a_{1} = 2$ and common difference $d = 3$. We use the formula:
\[a_{n} = a_{1} + ( n - 1 ) \cdot d\]Plug in the values:
\[a_{n} = 2 + ( n - 1 ) \cdot 3\]Now calculate the first few terms:
- $a_{1} = 2$
- $a_{2} = 2 + 1 \cdot 3 = 5$
- $a_{3} = 2 + 2 \cdot 3 = 8$
- $a_{4} = 2 + 3 \cdot 3 = 11$
- $a_{5} = 2 + 4 \cdot 3 = 14$
The resulting sequence is: \(2 , 5 , 8 , 11 , 14 , \ldots\)
Sum of $n$ terms of an arithmetic progression
The sum $S_{n}$ of the first $n$ terms $a_{1} , a_{2} , \ldots , a_{n}$ of an arithmetic progression is equal to the product of $n$ and the average of the first and last term:
\[S_{n} = n \cdot \frac{a_{1} + a_{n}}{2}\]This formula allows you to quickly compute the total sum of a finite number of terms in an arithmetic progression. For example, consider the arithmetic progression of non-negative even numbers: \(2 , 4 , 6 , 8 , 10\)
We want to calculate the sum of the first 5 terms $( n = 5 ) .$ Using the formula, we have: \(S_{5} = 5 \cdot \frac{2 + 10}{2} = 5 \cdot 6 = 30\)