Algebra 2
Unit 8 Day 6 – Series and Sequences
Name:_______________
Today:
 You will explore and apply properties of series and sequences of data.
 At the end of class you will be able to apply the properties of series and sequences to solve real world
problems and determine the nth term.
Warm Up:
1.
The table at right gives the year and [population (in millions)
of California. Determine the equation for the curve of best fit.
2.
Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will
be $6000.
Sequences and Series
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Vocabulary
Sequence: a function whose domain is a set of consecutive integers
Ex: 3, 6, 9, 12, 15
Domain: 1, 2, 3, 4, 5 (position of each term)
Range: 3, 6, 9, 12, 15 (terms of the sequence)
Finite sequence: a sequence whose last term exists
Infinite sequence: a sequence whose last term cannot be determined (. . .)
Rule: formula used to determine the terms of a sequence
Series: the sum of the terms of a sequence
Ex. 1: Write the first 6 terms of the sequence defined by the rule an  2n  3 .
Ex. 2: Write the first 6 terms of the sequence defined by the rule f (n)   2
n 1
5
Summation Notation (Sigma Notation)
 3i
lower limit:
upper limit:
i 1
Expanded form:
6
Ex. 3: Expand the series and find the sum:
 2i
i 1
6
Ex. 4: Evaluate:
 (2  k
2
)
k 3
Arithmetic Sequences and Series:
An arithmetic sequence is one whose consecutive terms have a common______________________.
For example: 1, 5, ____, _____, _____.
The common difference, d, is__________.
Ex. 1 Is the sequence arithmetic? If so, find the common difference and the next 2 terms.
5 11 7 1
1
a) 1, 4, 7, 10, 13, …
b) , , , ,  , ...
2 6 6 2
6
Arithmetic?_________
Arithmetic?_______
d? ______ next 2 terms:___________
d? _______ next 2 terms:________
Every Arithmetic Sequence has an nth term given by:
an  a1  (n  1)d
Where a1 is the
first term, d is the common difference, and n represents the number the term you are looking for.
Ex. 2 Find a formula for the nth term of the arithmetic sequence whose first term is 2 and whose common
difference is 5.
Ex. 3 Find the formula for the nth term of an arithmetic sequence if the common difference is 5 and the 2nd
term is 12.
Ex. 4 Find the 100th term in the sequence generated by an  4n  2
Ex. 5 Find the tenth term in the arithmetic sequence whose common difference is -3 and whose fourth term
is 1.
Ex. 6
-28 is the _______th term of 7, 2, -3, …
Geometric Sequences and Series:
A geometric sequence is one whose consecutive terms have a common ratio.
Ex. 1, 5, _____, _____, _____ The common ratio, r, is ____.
Ex. 7 Determining the common ratio
n
a) 12, 36, 108, 324, . . ., 4(3)n , . . .
r=_____
1 1
 1
b) 3, 1,  , ,...,9    ,...
3 9
 3
Every geometric sequence has an nth term of the form:
r=_____
an  a1r n 1
where a1 is the first term, n is the number of terms, and r is the common ratio of the sequence.
Ex. 8 Finding a term of a geometric sequence
a) Find the 20th term of the geometric sequence given a1  4 and r  2 .
b) Find the 11th term of the geometric sequence whose first 3 terms are 5, 15, and 45.
c) Find the number of terms of a geometric sequence with, first term 1/64, the common ratio 2, and the last
term 512.
Ex.9 Writing a rule for the nth term:
a) Write a rule for the nth term of the sequence 8,  12,  18,  27, . . .
b) One term of a geometric sequence is a3  5 . The common ratio is 2. Write a rule for the nth term.
c) Two terms of a geometric sequence are a2  45 and a5  1215 . Write a rule for the nth term.
Other Types of Sequences and Series:
Recursive: each term depends on the previous terms – look for a pattern!!
Ex. 10
Find the next three terms for the sequence: 5, 4, 9, 13, 22, …
Ex. 11 Find the next three terms for the sequence: 1, 1, 2, 3, 5, …
Ex. 12 Write the first six terms of the sequence given the following information:
a0  1, an  an1  4
Ex. 13 Write the first six terms of the sequence given the following information:
a1  1, an  3an1
Other Patterns….
Ex. 14 Find the first 4 terms of the sequence with the rule an  n 2
Ex. 15 Write a rule for the sequence:
1 4 9 16
, , ,
,...
2 3 4 5