Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, typically denoted by $d$.
Definition and Notation
For an arithmetic sequence $u_1, u_2, u_3, ...$, where $u_n$ represents the $n$th term:
$$u_n = u_{n-1} + d$$
where $d$ is the common difference.
ExampleConsider the sequence: 3, 7, 11, 15, 19, ...
Here, $a_1 = 3$ and the common difference $d = 4$.
NoteTerms can also be represented as $a_n$ instead of $u_n$. The two notations are interchangeable.
Formula for the nth Term
The general formula for the $n$th term of an arithmetic sequence is:
$$u_n = u_1 + (n-1)d$$
where $u_1$ is the first term, $n$ is the position of the term, and $d$ is the common difference.
This is because for every term, you're adding $d$ to the previous term. To get to the $n$th term from the first term, you perform $n - 1$ additions of $d$.
TipTo find the common difference quickly, subtract any term from the subsequent term: $d = a_{n+1} - a_n$
ExampleFor the sequence 3, 7, 11, 15, 19, ..., find the 10th term.
Solution: $a_1 = 3$, $d = 4$ $a_{10} = 3 + (10-1)4 = 3 + 36 = 39$
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. This is denoted by $S_n$, where $n$ is the number of terms you are summing together.
Formulas for arithmetic series
Let's take the first and last terms of an arithmetic sequence, or $u_1$ and $u_n$. The sum of these is $(u_1 + u_n)$.
Now, let's take the second and second-to-last terms of an arithmetic sequence, or $u_2$ and $u_{n-1}$. These can be expressed as $u_1 + d$ and $u_n - d$ respectively.
Therefore, $u_2 + u_{n-1} = u_1 + d + u_n - d = u_1 + u_n$.
