UNIT 12: SERIES AND SEQUENCES
Sequence Definitions

A sequence is an ordered list of numbers. example: 1, 4, 7, 10, 13

The sum of the terms in a sequence is called a series. example: 1 + 4 + 7 + 10 + 13

Each number of a sequence is called a term.

A finite sequence contains a finite number of terms. example: 1, 4, 7, 10, 13

An infinite sequence contains an infinite number of terms. example: 1, 4, 7, 10, 13, ….

The terms of a sequence are referred to in the subscripted form shown below, where the
subscript refers to the location (position) of the term in the sequence.
Arithmetic Sequences

If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it
is referred to as an arithmetic sequence. The number added to each term is constant (always
the same).

The fixed amount is called the common difference, d, referring to the fact that the difference
between two successive terms yields the constant value that was added.

To find the common difference, subtract the first term from the second term:
Arithmetic Sequence Formula (Rule) {On reference sheet}

(
)
To find any term of an arithmetic sequence use the formula
where a1 is the first term of the sequence, d is the common difference, n is the number of the term
to find.

To write the formula/rule of any arithmetic sequence, all you need is the 1st term and the
common difference
Geometric Sequences

If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term
to arrive at the following term, it is referred to as a geometric sequence. The number multiplied
each time is constant (always the same).

The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio
(fraction) of the second term to the first term yields this common multiple.

To find the common ratio, divide the second term by the first term:
Writing the Formula/Rule of a Geometric Sequence {On reference sheet}


To find any term of a geometric sequence:
where a1 is the first term of the sequence, r is the common ratio, n is the number of the term to
find.
To write the formula/rule of any geometric sequence, all you need is the 1st term and the
common ratio
Recursive Sequences
Recursion is the process of choosing a starting term and repeatedly applying the same process to each
term to arrive at the following term. Recursion requires that you know the value of the term
immediately before the term you are trying to find.
A recursive formula always has two parts:
1. the starting value for a1.
2. the recursion equation for an as a function of an-1 (the term before it.)
Recursive Sequences (Continued)

Recursion notation comes in many different forms.
*For multiple choice questions, you need to know that all of these mean the same thing!
Converting from Recursive to Explicit
Use the following explicit formulas with
as first term (Use for
) (GIVEN)
**These formulas are given to you on the reference sheet!**
ARITHMETIC SEQUENCE:
(
GEOMETRIC SEQUENCE:
)
Since you are adding 2 to the previous term, this represents an arithmetic sequence.
(
) ; where
Therefore, the explicit form of this recursive sequence is:
(
)( )
Convert from Explicit to Recursive
Use the following recursive formulas (MEMORIZE)
**Remember to include the first term or terms when writing the recursive formula!**
ARITHMETIC SEQUENCE:
GEOMETRIC SEQUENCE:
Subscript notation:
Subscript notation:
and
( )
Function notation:
and ( )
(
)
( )
Below you are given the explicit form for the sequence ( ).
Therefore, the recursive sequence for ( ) is:
( )
( )
(
)
Function notation:
and ( )
(
)
Summation Notation
Mathematicians use notation to reduce the amount of writing and to prevent ambiguity in mathematical
sentences. Mathematicians use a special symbol with sequences to indicate that we would like to sum
up the sequence. This symbol is a capital sigma, . The sum of a sequence is called a series.

The summation of a specific number of terms of a sequence (a series) can also be represented in
a compact form, called summation notation, or sigma notation.

The Greek capital letter sigma,

Given a sequence
using the expression:
, is used to indicate a sum.
we can write the sum of the first
terms of the sequence
∑

It is read, “The sum of

Example:

from
to
.”
To write the terms of the series, replace n by the consecutive integers from 1 to 5, as shown
above.
To Find the Summation Button On The Calculator:
Math
0: Summation ∑(
Use Alpha Key to get the variables
**Remember: The entire expression must be within the given parenthesis to correctly find the sum.
**GEOMETRIC SERIES** (on Reference Sheet!)
To find the sum of a certain number of terms of a geometric sequence:
where Sn is the sum of n terms,
a1 is the first term,
r is the common ratio.

Example 1: Find the following sum:
Solution: This is a geometric series with the first term
first 8 terms. Therefore, we have:
and the common ratio
. We are adding up the
( )

Example 2: Find the following sum:
.
Solution: This is a geometric series with the first term
and the common ratio
. We do not know how
many terms we have (n). Therefore, we must use the geometric sequence formula to find the value of n.
( )
where
( )
( )
( )
Now we use the geometric formula to find the sum of the six terms:
( )