Section 11.2

Arithmetic Sequences

Arithmetic sequences are the most straightforward type on Exam 4, and they're great warm-up for the harder geometric and series problems. The sum formula also comes back when you do summation notation in section 9-3.

⚡ Quick Summary

An arithmetic sequence is the simplest kind of pattern: you pick a starting number and keep adding the same amount every time. Think of climbing stairs — each step takes you the same height higher. The number you add each time is called the common difference $d$ . If your sequence is 5, 8, 11, 14, ... then $d = 3$ because you're adding 3 every step.

Explicit (nth term) formula

a_n = a_1 + (n - 1)d

Jumps directly to the $n$ th term without computing every term before it.

Recursive formula

a_n = a_{n-1} + d

Defines each term from the previous one; you also need the first term $a_1$ .

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The formula $a_n = a_1 + (n-1)d$ is a shortcut to jump to any term without listing them all. Read it as: "start at $a_1$ , then take $(n-1)$ jumps of size $d$ ." Why $(n-1)$ and not $n$ ? Because the first term is your starting point — you haven't jumped yet! So the 10th term means 9 jumps from the start.

When you add up all the terms in an arithmetic sequence, the trick is simple: the sum equals the average of the first and last term, multiplied by how many terms there are. That's $S_n = \frac{n}{2}(a_1 + a_n)$ . The legend goes that the mathematician Gauss figured this out as a child when his teacher told the class to add the numbers 1 through 100 — he paired 1+100, 2+99, 3+98, and so on, getting 50 pairs of 101 each, for a total of 5050.

If a problem gives you two terms (like the 4th and 10th), you can find $d$ by writing two equations from the formula and subtracting them. The $a_1$ terms cancel, and you're left with a simple equation for $d$ . Then plug $d$ back in to find $a_1$ . Real-world examples of arithmetic sequences include seat counts in stadium rows, stacking patterns, and regular savings deposits.

Key Formulas

Explicit (nth term) formula

a_n = a_1 + (n - 1)d

Jumps directly to the $n$ th term without computing every term before it.

Recursive formula

a_n = a_{n-1} + d

Defines each term from the previous one; you also need the first term $a_1$ .

Sum of an arithmetic series (version 1)

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

Use when you know both the first and last term.

Sum of an arithmetic series (version 2)

S_n = \frac{n}{2}\bigl(2a_1 + (n - 1)d\bigr)

Use when you know $a_1$ , $d$ , and $n$ but not $a_n$ .

Key Takeaways

✓Explicit formula: $a_n = a_1 + (n-1)d$ — note it's $(n-1)$ , not $n$ , because $a_1$ gets $d$ added zero times
✓Sum formula: $S_n = \frac{n}{2}(a_1 + a_n)$ — it's the average of first and last, times the count
✓Given two terms, subtract the equations to find $d$ — then back-substitute for $a_1$
✓If terms go down by a constant, $d$ is negative — the formulas still work the same way

⚠️ Common Mistakes

✗Using $n$ instead of $(n-1)$ in the explicit formula — $a_n = a_1 + (n-1)d$ , not $a_n = a_1 + nd$ ; the first term gets $d$ added zero times
✗When given two terms like $a_4$ and $a_{10}$ , trying to guess $d$ instead of setting up two equations and subtracting to solve for $d$ systematically
✗Mixing up the two sum formulas — use $S_n = \frac{n}{2}(a_1 + a_n)$ when you know the last term; use $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ when you don't
✗Forgetting that $d$ can be negative — if terms decrease, $d < 0$ , but all the formulas still work exactly the same way

Next: Section 9.3 Builds on: 11.1

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11.1 Sequences & Notation

11.3 Geometric Sequences