13.2 Arithmetic Sequences

Arithmetic Sequences

Arithmetic sequences are lists of numbers where the gap between consecutive terms is always the same. That constant gap is called the common difference, and once you know it along with the first term, you can find any term in the sequence. These sequences model situations with steady, constant change, like adding the same amount to a savings account each month or scheduling equal time intervals.

Common Difference

The common difference ($d$) is the value added to each term to get the next one. You calculate it by subtracting any term from the term that follows it:

$d = a_{n+1} - a_n$

• If $d > 0$, the sequence increases (e.g., 2, 5, 8, 11, ...)
• If $d < 0$, the sequence decreases (e.g., 20, 17, 14, 11, ...)
• If $d = 0$, every term is the same (e.g., 4, 4, 4, 4, ...)

The common difference stays the same for the entire sequence. If the difference between terms changes at any point, it's not arithmetic.

Generating Terms

An arithmetic sequence is fully defined by its first term $a_1$ and its common difference $d$. To build the sequence, start at $a_1$ and keep adding $d$:

• $a_2 = a_1 + d$
• $a_3 = a_1 + 2d$
• $a_4 = a_1 + 3d$

Notice the pattern: the coefficient in front of $d$ is always one less than the term number. This observation leads directly to the explicit formula covered below.

Recursive Formula

A recursive formula defines each term using the previous term:

$a_{n+1} = a_n + d$

To use it, you need two pieces of information: the first term $a_1$ and the common difference $d$. Then you build the sequence one step at a time.

Example: $a_1 = 3$, $d = 2$, so the recursive formula is $a_{n+1} = a_n + 2$.

1. $a_2 = 3 + 2 = 5$
2. $a_3 = 5 + 2 = 7$
3. $a_4 = 7 + 2 = 9$

The recursive approach is straightforward, but it has a drawback: to find the 50th term, you'd need to calculate all 49 terms before it. That's where the explicit formula comes in.

Explicit Formula

The explicit formula lets you jump straight to any term without computing all the ones before it:

$a_n = a_1 + (n - 1)d$

• $a_n$ = the term you're looking for
• $a_1$ = the first term
• $n$ = the position of the term
• $d$ = the common difference

Example: Find the 10th term when $a_1 = 3$ and $d = 2$.

1. Plug into the formula: $a_{10} = 3 + (10 - 1)(2)$

2. Simplify inside the parentheses: $a_{10} = 3 + (9)(2)$

3. Multiply and add: $a_{10} = 3 + 18 = 21$

This formula is especially useful when you need a term far into the sequence, like the 200th term, without grinding through every previous one.

Sequences vs. Series

A sequence is an ordered list of numbers that follows a pattern. An arithmetic sequence is the specific case where consecutive terms differ by a constant amount.

A series is what you get when you add up the terms of a sequence. So while the sequence 2, 5, 8, 11 is a list, the series $2 + 5 + 8 + 11 = 26$ is a sum. Keep these terms straight, since exam questions often test whether you know the difference.