Algebra II/Trig Honors
Unit 7 Day 1: Define and Use Sequences and Series
Objective: To recognize and write rules for number patterns
Key Terms:

Sequence - ______________________________________________________________
________________________________________________________________________
o If the domain is not specified, _________________________________________.
o The values in the range are called the ___________________________________.
Domain: 1
2
3
4 . . . n
The position of each term
Range: a1
a2
a3
a4 . . . a n
Terms of the sequence

Finite sequence - _______________________________________________________
o Ex. 2, 4, 6, 8

Infinite sequence - ______________________________________________________
o Ex. 2, 4, 6, 8, …

A sequence can be specified by an equation, or rule. Both sequences above can be
described by the rule a n  2n or f n  2n .
Example 1: Write terms of sequences
Write the first six terms of each sequence.
a. an  2n  5
b. f n    3
n 1
Writing Rules - If the terms of a sequence have a recognizable pattern, then you may be able to
write a rule for the n-th term of the sequence.
Example 2: Write rules for sequences
Describe the pattern, write the next term, and write a rule for the n-th term of the sequences.
a. -1, -8, -27, -64, . . . . .
b. 0, 2, 6, 12, . . . . .
Graphing Sequences – To graph a sequence, let the horizontal axis represent the position
numbers (the domain) and the vertical axis represent the terms (the range).
Example 3: Solve a multi-step problem
You work in a grocery store and are stacking apples in the shape of a square pyramid with 7
layers. Write a rule for the number of apples in each layer. Then graph the sequence. (Let the top
of the pyramid be the first layer).
More Key Terms:

Series - ________________________________________________________________
o Finite series: 2 + 4 + 6 + 8
o Infinite series: 2 + 4 + 6 + 8 + . . . .

Summation Notation - ___________________________________________________

4
Finite Series: 2 + 4 + 6 + 8 =
 2i
Infinite Series: 2 + 4 + 6 + 8 + . . . =
i 1
 2i
i 1
Read as “the sum of 2i for values of i from 1 to 4”
o The index of the summation is i (can be any letter – most common: i, k, and n)
o The lower limit of summation is 1
o The upper limit of summation is 4 for the finite series,  for the infinite series
Example 4: Write series using summation notation
Write the series using summation notation.
a. 25 + 50 + 75 + . . . + 250
b.
1 2 3 4
    ....
2 3 4 5
Example 5: Find the sum of a series
 3  k 
8
Find the sum of the series
2
k 4
Special Formulas for Special Series
n
Sum of n terms of a constant:
 c  cn
i 1
n
Sum of first n positive integers:
i 
i 1
nn  1
2
n
Sum of the squares of first n positive integers:
i
i 1
2

nn  12n  1
6
Example 6: Use a formula for a sum
How many apples are in the stack in Example 3?
HW: Page 438 #4-56 (M4), 58-62