Unit 9: Sequences and Series
Sequences
 A sequence is a list of #s in a particular order
 If the sequence of numbers does not end, then it is
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called an infinite sequence
Each # in a sequence is called a term
Ex. 3, 5, 7, 9….
a1=3
a2=5
Arithmetic Sequences
 A sequence in which each term after the first term is
found by adding a constant (called the common
difference (d)), to the previous term
 Ex. Find the common difference (one term minus
the previous term)
 55, 49, 43, 37, 31, 25,19
Is the sequence arithmetic? If so give the
common difference.
 -4, -1, 2, 5, 8……
 7, 3, -1, -5, -9……
 3, 6, 12, 24……
 3, 8
Use the recursive formula to find the first four
terms of the arithmetic sequence given.
 𝑡1 = 7
 𝑡𝑛 = 𝑡𝑛−1 − 3
Use the recursive formula to write the first four
terms of the arithmetic sequence.
 𝑡1 = −2
 𝑡𝑛 = 𝑡𝑛−1 + 6
List the first three terms of the arithmetic
sequence.
 𝑡𝑛 = 2𝑛 − 3
Another one. . .
 𝑡𝑛 = 5 + 𝑛 − 1 (3)
Let’s figure out the explicit formula.
 1st Term:
 2nd Term:
 3rd Term:
 4th Term:
 10th Term:
 nth Term:
Arithmetic Sequences
 Formula for the general pattern for any
arithmetic sequence:
an  a1  (n  1)d
where...
an  value of the n term
th
a1  value of the1st term
n # of terms
d  common difference
Arithmetic Sequences
an  a1  (n  1)d
 Write a equation for the nth term of the
following sequence
8,17, 26, 35, ...
Arithmetic Sequences
an  a1  (n  1)d
 Write a equation for the nth term of the
following sequence
5 7
2, , 3, , ...
2 2
Arithmetic Sequences
 Find a15
9,7,5,3,...
an  a1  (n  1)d
Arithmetic Sequences
an  a1  (n  1)d
 In the sequence below, which term has a value
of 286?
2, 6,10,14, ...
Arithmetic Sequences
an  a1  (n  1)d
 What is the value of the first term if the 9th and
10th terms are 4 and 2 consecutively?
Arithmetic Sequences
an  a1  (n  1)d
 If the 3rd term of an arithmetic sequence is 8
and the 16th term is 47, find a1 and d
Arithmetic Means
an  a1  (n  1)d
 Terms between any two non-successive terms
of an arithmetic sequence
 Find 4 arithmetic means between 16 and 91
Arithmetic Means
an  a1  (n  1)d
 Find 1 arithmetic mean between 50 and -120
Arithmetic Series
 A series is the indicated sum of the terms of a
sequence
 If the seq. is 18, 22, 26, 30
 The arith series is 18 + 22 + 26 + 30
Arithmetic Series
 To find the sum of an
arithmetic series,
n
S n  a1  an 
2
Arithmetic Series
 Find the sum of the
first 100 positive
integers.
n
S n  a1  an 
2
Arithmetic Series
 Find the sum of the
first 20 even numbers
beginning with 2.
n
S n  a1  an 
2
Arithmetic Series
 Find the sum of
34+30+26+…+2
n
S n  a1  an 
2
Arithmetic Series
 Find the sum of
6+13+20+27…+97
n
S n  a1  an 
2