Sequences and Series: Arithmetic, Geometric, and Beyond

2026.04.14

Learn.byAuthor

A sequence is an ordered list of numbers following a pattern. A series is what you get when you add them up. These concepts are fundamental to calculus and appear across standardized tests.

💡

Key Point: Sequences and series appear on AP Calculus BC (convergence tests), A-levels, and college entrance exams. Arithmetic and geometric types cover 90% of exam questions.

Arithmetic Sequences

Each term increases by a constant difference $d$ .

a_n = a_1 + (n-1)d

$a_1$ : first term
$d$ : common difference
$a_n$ : nth term

Example: $2, 5, 8, 11, \ldots$ has $a_1 = 2$ , $d = 3$ .

Sum of an Arithmetic Series

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

Geometric Sequences

Each term is multiplied by a constant ratio $r$ .

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

$a_1$ : first term
$r$ : common ratio
$a_n$ : nth term

Example: $3, 6, 12, 24, \ldots$ has $a_1 = 3$ , $r = 2$ .

Sum of a Geometric Series

Finite ( $r \neq 1$ ):

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

Infinite ( $|r| < 1$ ):

S_{\infty} = \frac{a_1}{1 - r}

⚠️

Watch Out: The infinite sum formula only works when

|r| < 1

. If

|r| \geq 1

, the series diverges and has no finite sum.

Comparison Table

Property	Arithmetic	Geometric
Pattern	Add $d$	Multiply by $r$
nth term	$a_1 + (n-1)d$	$a_1 \cdot r^{n-1}$
Finite sum	$\frac{n}{2}(a_1 + a_n)$	$a_1 \cdot \frac{1-r^n}{1-r}$
Infinite sum	Diverges (always)	$\frac{a_1}{1-r}$ if $\|r\| < 1$

How to Identify the Type

Check the differences: Subtract consecutive terms. If they’re constant → Arithmetic.
Check the ratios: Divide consecutive terms. If they’re constant → Geometric.
Neither constant? → It might be a different type (quadratic, recursive, etc.).

Worked Examples

Example 1: Arithmetic — Find the 20th Term

For the sequence $5, 9, 13, 17, \ldots$ , find $a_{20}$ .

Show Solution

$a_1 = 5$ , $d = 4$

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

Example 2: Geometric — Sum of First 6 Terms

For $a_1 = 2$ , $r = 3$ , find $S_6$ .

Show Solution

S_6 = 2 \cdot \frac{1 - 3^6}{1 - 3} = 2 \cdot \frac{1 - 729}{-2} = 728

Example 3: Infinite Geometric Series

Find the sum of $8 + 4 + 2 + 1 + \ldots$

Show Solution

$a_1 = 8$ , $r = \frac{1}{2}$ . Since $|r| < 1$ :

S_{\infty} = \frac{8}{1 - \frac{1}{2}} = 16

📝

Pro Tip: When given a series problem, always identify whether it’s arithmetic or geometric first. This determines which formula to use. Check differences, then ratios.

Top 3 Common Mistakes

Off-by-one error in the nth term — it’s $(n-1)d$ , not $nd$ , because the first term has zero differences added
Using the infinite sum when $|r| \geq 1$ — the series diverges, so there’s no finite answer
Confusing $a_n$ with $S_n$ — $a_n$ is a single term, $S_n$ is the sum of the first $n$ terms

Sequences and Series: Arithmetic, Geometric, and Beyond

Arithmetic Sequences

Sum of an Arithmetic Series

Geometric Sequences

Sum of a Geometric Series

Comparison Table

How to Identify the Type

Worked Examples

Example 1: Arithmetic — Find the 20th Term

Example 2: Geometric — Sum of First 6 Terms

Example 3: Infinite Geometric Series

Top 3 Common Mistakes

Related Topics