12.2 Arithmetic Sequences

Arithmetic Sequences

An arithmetic sequence is a list of numbers where you always add (or subtract) the same value to get from one term to the next. That fixed value is called the common difference. Understanding arithmetic sequences lets you predict any term in a pattern and quickly calculate the sum of many terms, which shows up in finance, physics, and plenty of algebra problems.

Patterns in arithmetic sequences

To check whether a sequence is arithmetic, subtract any term from the one that follows it. If that difference is the same every time, you've got an arithmetic sequence. The constant difference is called the common difference, written as $d$.

• Positive common difference: 2, 5, 8, 11, 14, ... Here $d = 3$ because each term is 3 more than the previous one. The sequence increases.
• Negative common difference: 10, 7, 4, 1, -2, ... Here $d = -3$ because each term is 3 less than the previous one. The sequence decreases.

If the differences between consecutive terms aren't all equal, the sequence is not arithmetic. For example, 1, 2, 4, 8, ... has differences of 1, 2, 4, so it fails the test.

General term formula construction

Instead of listing out every term, you can jump straight to any term using the general term formula:

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

• $a_n$ = the term you want to find (the $n$th term)
• $a_1$ = the first term
• $n$ = the position of the term in the sequence
• $d$ = the common difference

How to build the formula:

1. Identify the first term $a_1$.

2. Find the common difference $d$ by subtracting any term from the next one.

3. Plug $a_1$ and $d$ into $a_n = a_1 + (n - 1)d$.

4. Simplify if needed.

Example 1: For the sequence 2, 5, 8, 11, 14, ...

• $a_1 = 2$, $d = 5 - 2 = 3$
• $a_n = 2 + (n - 1)(3) = 2 + 3n - 3 = 3n - 1$
• Quick check: $a_4 = 3(4) - 1 = 11$ ✓

Example 2: For the sequence 10, 7, 4, 1, -2, ...

• $a_1 = 10$, $d = 7 - 10 = -3$
• $a_n = 10 + (n - 1)(-3) = 10 - 3n + 3 = 13 - 3n$
• Quick check: $a_5 = 13 - 3(5) = -2$ ✓

Always do a quick check like this by plugging in a position you already know. It catches sign errors fast.

Sum of finite arithmetic sequences

When you need the sum of the first $n$ terms of an arithmetic sequence, use this formula:

$S_n = \frac{n}{2}(a_1 + a_n)$

• $S_n$ = sum of the first $n$ terms
• $a_1$ = first term
• $a_n$ = the $n$th (last) term you're summing to
• $n$ = number of terms

The idea behind this formula: if you pair the first term with the last, the second with the second-to-last, and so on, each pair has the same sum. There are $\frac{n}{2}$ such pairs, each totaling $a_1 + a_n$.

Steps to find the sum:

1. Identify $a_1$, $d$, and $n$.

2. If you don't already know $a_n$, calculate it using $a_n = a_1 + (n - 1)d$.

3. Plug $a_1$, $a_n$, and $n$ into $S_n = \frac{n}{2}(a_1 + a_n)$.

Example 1: Find the sum of the first 10 terms of 2, 5, 8, 11, 14, ...

• $a_1 = 2$, $d = 3$, $n = 10$
• $a_{10} = 2 + (10 - 1)(3) = 2 + 27 = 29$
• $S_{10} = \frac{10}{2}(2 + 29) = 5(31) = 155$

Example 2: Find the sum of the first 6 terms of 10, 7, 4, 1, -2, ...

• $a_1 = 10$, $d = -3$, $n = 6$
• $a_6 = 10 + (6 - 1)(-3) = 10 - 15 = -5$
• $S_6 = \frac{6}{2}(10 + (-5)) = 3(5) = 15$

Types of Sequences and Series

A sequence is an ordered list of numbers. A series is what you get when you add the terms of a sequence together.

• A finite sequence has a definite number of terms (like the first 10 terms of a pattern). You can always find its sum.
• An infinite sequence continues without end (indicated by "..." at the end). You can still write a general term formula for it, but the sum formula $S_n$ only applies when you pick a specific number of terms to add.

Arithmetic sequences can be described either way depending on context. If a problem says "find the sum of the first 20 terms," you're treating it as finite even if the pattern itself could go on forever.