🔢 Sequences and Series Basics

Common Examples

• Even numbers: 2, 4, 6, 8, 10, ...
• Odd numbers: 1, 3, 5, 7, 9, ...
• Powers of 2: 1, 2, 4, 8, 16, 32, ...
• Squares: 1, 4, 9, 16, 25, ...
• Fibonacci: 1, 1, 2, 3, 5, 8, 13, ...

Notation

We use subscripts to denote position:

• $a_1$ = first term
• $a_2$ = second term
• $a_3$ = third term
• $a_n$ = $n$th term (general term)

Example: In the sequence 3, 7, 11, 15, ...

$a_1 = 3$, $a_2 = 7$, $a_3 = 11$, $a_4 = 15$

Definition

Each term is the previous term plus a constant $d$ (common difference)

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

Example 1: Find the 20th term

Sequence: 5, 9, 13, 17, 21, ...

• $a_1 = 5$ (first term)
• $d = 4$ (common difference)

Find $a_{20}$:

$a_{20} = 5 + (20 - 1)(4) = 5 + 76 = 81$ ✨

Example 2: Is 87 in this sequence?

Sequence: 3, 7, 11, 15, ... (where $d = 4$)

Set $a_n = 87$:

$87 = 3 + (n - 1)(4)$

$84 = (n - 1)(4)$

$n - 1 = 21$ → $n = 22$ ✓

Yes! 87 is the 22nd term.

Finding the Common Difference

Simply subtract consecutive terms!

For 2, 5, 8, 11: $d = 5 - 2 = 3$

Definition

Each term is the previous term times a constant $r$ (common ratio)

$$a_n = a_1 \cdot r^{n-1}$$

Example 1: Find the 6th term

Sequence: 3, 6, 12, 24, 48, ...

• $a_1 = 3$ (first term)
• $r = 2$ (common ratio)

Find $a_6$:

$a_6 = 3 \cdot 2^{6-1} = 3 \cdot 32 = 96$ ✨

Example 2: Decay Sequence

Sequence: 100, 50, 25, 12.5, ...

• $a_1 = 100$
• $r = 0.5$ (or $\frac{1}{2}$)

Find $a_5$:

$a_5 = 100 \cdot (0.5)^4 = 100 \cdot 0.0625 = 6.25$

Finding the Common Ratio

Divide consecutive terms!

For 5, 15, 45, 135: $r = \frac{15}{5} = 3$

Formula for Arithmetic Series

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

Or equivalently:

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Famous Example: Sum 1 + 2 + 3 + ... + 100

This is an arithmetic series with:

• $n = 100$ (100 terms)
• $a_1 = 1$, $a_{100} = 100$

$S_{100} = \frac{100(1 + 100)}{2} = \frac{100 \cdot 101}{2} = 5050$ ✨

Example: Find the sum of the first 20 terms

Sequence: 5, 9, 13, 17, ... ($d = 4$)

First, find $a_{20}$: $a_{20} = 5 + 19(4) = 81$

Now find the sum:

$S_{20} = \frac{20(5 + 81)}{2} = 10 \cdot 86 = 860$

Formula for Geometric Series

$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$

(when $r \neq 1$)

Example 1: Sum the first 5 terms

Sequence: 2, 6, 18, 54, ...

• $a_1 = 2$
• $r = 3$
• $n = 5$

$S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{1 - 243}{-2}$

$= 2 \cdot \frac{-242}{-2} = 2 \cdot 121 = 242$ ✨

Example 2: Sum with fraction ratio

Find: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}$

$a_1 = 1$, $r = \frac{1}{2}$, $n = 5$

$S_5 = 1 \cdot \frac{1 - (\frac{1}{2})^5}{1 - \frac{1}{2}} = \frac{1 - \frac{1}{32}}{\frac{1}{2}}$

$= \frac{\frac{31}{32}}{\frac{1}{2}} = \frac{31}{16}$

Quick Test Method

Check differences: If constant → Arithmetic

Check ratios: If constant → Geometric

Example 1: 2, 5, 8, 11, 14

Differences: $5-2=3$, $8-5=3$, $11-8=3$ ✓

Arithmetic with $d = 3$

Example 2: 3, 9, 27, 81

Ratios: $\frac{9}{3}=3$, $\frac{27}{9}=3$, $\frac{81}{27}=3$ ✓

Geometric with $r = 3$

Example 3: 1, 4, 9, 16, 25

Differences: 3, 5, 7, 9 (not constant)

Ratios: 4, 2.25, 1.78... (not constant)

Neither! (These are perfect squares)