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.
Example- Consider the sequence: 3, 7, 11, 15, 19, ...
- Here, $a_1 = 3$ and the common difference $d = 4$.
- Terms 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
- $d$ 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 $d$.
To find the common difference quickly, subtract any term from the subsequent term: $d = a_{n+1} - a_n$
Example question
For the sequence 3, 7, 11, 15, 19, ..., find the 10th term.
Solution
$$a_1 = 3 \qquad d = 4 $$
$$a_{10} = 3 + (10-1)4 = 3 + 36 = 39$$
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 $u_n$.
