Arithmetic & Geometric Sequences

Sequences are ordered lists of numbers created by following a pattern.
Two of the most important types of sequences in Algebra 2, Precalculus, and Calculus BC foundations are:

• Arithmetic sequences

• Geometric sequences

These appear in growth models, series, and early steps of infinite sum problems.

What Is a Sequence?

A sequence is a list of numbers written in order:

$$a_{1} , a_{2} , a_{3} , \dots$$

Each number is called a term.
A formula for the nth term is called a general term or explicit formula.

Arithmetic Sequences

A sequence is arithmetic when the difference between consecutive terms is constant.

This constant is called the common difference:

$$d=a_{n+1}-a_{n}$$​

Examples:

sequence $$3 , 7 , 11 , 15 , \dots$$ , common difference d = 7 – 3 = 4

sequence $$20 , 17 , 14 , 11 , \dots$$ , common difference d = 17 – 20 = -3

general term (nth Term)

$$a_{n}=a_{1}+(n+1)d$$​

This gives any term directly.

Example

Arithmetic sequence: 5, 9, 13, 17, …
Here, $$a_{1}=first term=5 , common difference d = 9 – 5 = 4$$

$$a_{n}=5+(n-1)\cdot4=4n+1$$

Geometric Sequences

A sequence is geometric when the ratio between consecutive terms is constant.

This constant is the common ratio:

$$r=\frac{a_{n+1}}{a_{n}}$$​​

Examples:

sequence $$3 , 6 , 12 , 24 , \dots$$ common ratio r =$$\frac{6}{3}=2$$

sequence $$81 , 27 , 9 , 3 , \dots$$ common ratio r =$$\frac{27}{81}=\frac{1}{3}$$

general term (nth Term)

$$a_{n}=a_{1}r^{n-1}$$​

Example

Sequence: 2, 8, 32, 128, … $$a_{1}=2 common ratio r = \frac{8}{2}=4$$

$$a_{n}=2\cdot4^{n-1}$$​

Story Applications (Growth Models)

Arithmetic sequences model linear growth

Example: You save $20 each week → arithmetic pattern.

Geometric sequences model exponential growth/decay

Example: Population increases 5% per year, or radioactive decay → geometric pattern.