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Day 1: Introduction to Sequences
Alg 2X Unit 10 Sequences and Series
Example: 1, 3, 5, 7, 9, 11, …
A _____________________ is an ordered set of numbers.
Each number is called a _________________ of the sequence.
What is the 1st term? _______
3rd term? _________
6th term? ________
10th term? __________
The sequence above is called an _______________ sequence because it goes on forever (notice the …).
If the sequence ends (ex: 2, 4, 6, 8), then it is called a _______________ sequence.
Sometimes it is helpful to list your sequence in a table to help you map the number of the term to the term itself.
Term Number
Term Value
Normally, we will use sequence notation: n and an .
n represents _____________________ and an represents _____________________
So for example, a4  7 means “The _________ term is equal to ________”
Examples: Use the sequence :
a) a2  _____
2, 6, 10, 14, 18, 22, 26
b) a5  _____
c) a___  10
d) a___  14
d) a7  _____
Ok, now that we understand the notation, let’s talk a little more abstractly about sequences.
What is the term previous to an ? __________.
What is the next term after an ? ___________.
What is the next term after an 3 ? ___________.
What is the term previous to an 3 ? ___________.
RECURSIVE FORMULA vs EXPLICIT FORMULA:
____________ Formula – _____________________________ (MUST identify one term; often includes “an-1”)
____________ Formula – _____________________________ (often includes “n”)
EXAMPLE:
3, 6, 9, 12, 15, …
RECURSIVE FORMULA:
EXPLICIT FORMULA:
Recursive Formula Problems:
1. a1 = 5 and an  2(an1 )  3
Find the first 5 terms in the sequence.
2. Find the first 5 terms in the sequence with a1 = - 2 and an  3(an1 )  2
3. Find the first 5 terms in the sequence which has a1  7 and an  2(an1 )  4
4. What is your first thought if I ask you to find the 1000th term of the sequence in #3?
Another way to explain EXPLICIT FORMULA: An explicit formula defines the nth term in a sequence as a
function of n.
Example: Find the first 5 terms of the sequence an  2(n  5)  3 .
Find the 1000th term of the same sequence:
Explicit Formula Problems:
1. Find the first 5 terms of the sequence an  n 2  2n . Then find the 25th term and the 100th term.
2. Find the first 5 terms of the sequence an  3n  5 .
Then find a25 and a100.
3. Find the 3rd, 6th, 10th and 17th terms in the sequence an  3n  1
FINDING FORMULAS: Find an explicit formula for each sequence: You may use the table to help.
1.
3, 6, 9, 12, 15, 18, …
Term Number
Term Value
Formula:
an =
2.
1 2 3 4 5
, , , ,
2 3 4 5 6
Term Number
Term Value
Formula:
3.
an =
1, 4, 9, 16, 25, 36 …
Term Number
Term Value
Formula:
4.
an =
4, 9, 16, 25, 36, 49 …
Term Number
Term Value
Formula:
an =
Quick Wrap-Up: Find a5 for each sequence below:
a) a1  3 ; an  an1  4
b) an  8n  7