**Introduction to Arithmetic Sequences**

**What is an Arithmetic Sequence?**

A sequence is just a list of numbers.

For example: 1, 2, 3, 4, 5, 6… is a sequence. We refer to a sequence by its terms. For this example, we can see that we simply add 1 to the previous term in the list to get the next term.

a_{n}= a_{n-1}+ 1

ais the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations._{n}

a_{1}is the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula.

nis treated like the variable in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be 12.

dis the common difference for the arithmetic sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for d into the formula.

20, 24, 28, 32, 36, . . .

Sometimes we will need to know if a given value is a term in a sequence.

Is 623 a term in the sequence 4, 10, 16, 22, . . . ?

First we need to find the explicit formula for the sequence.

**Khan Academy**

*Intro to arithmetic sequences**Intro to arithmetic sequence formulas**Recursive and Explicit formulas*

**Introduction to Geometric Sequences**

**What is a Geometric sequence?**2, 6, 18, 54, 162, . . .

ais the nth term of the sequence. When writing the general expression for a geometric sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations._{n}ais the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula._{1 }ris the common ratio for the geometric sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for r into the formula.

**Khan Academy**

*Intro to geometric sequences (video)**Geometric sequence formulas (video)**Recursive and explicit*

Worksheet: Arithmetic and Geometric Sequences – practice

(*Thank you to Algebra LAB for their examples and definitions of recursive and explicit formulas)