Section 12.1
Sequences and Series
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; unless stated otherwise starts from 1)
Range: 3, 6, 9, 12, 15 (terms of the sequence)
 Notation: use _____ instead of _______ and _______ instead of ________


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
o Explicit rule
o Recursive rule
 Series: the sum of the terms of a sequence (can also be finite or infinite)
Ex. 1: Write the first 6 terms of the sequence defined by the rule a n  2n  3 .
Ex. 2: Write the first 6 terms of the sequence defined by the rule f (n)   2
n 1
Ex. 3: For given sequence describe the pattern, write the next term, and write the rule for the nth term
a) 1, 4, 9, 16…
b) -3, 9, -27, 81…
c) 0, 7, 26, 63…
5
Summation Notation (Sigma Notation)
 3i
Rule (explicit) / function
i 1
Index is used to tell the rule;
 Lower limit:
 Upper limit:
Expanded form:
6
Ex. 4: Expand the series and find the sum:
 2i
i 1
6
Ex. 5: Evaluate:
 (2  k
2
)
k 3
Ex. 6: Write the given series using summation notation:
a) 1 + 4 + 9 + 16…
Rule
Lower limit
Upper limit
b) 4 + 8 + 12 + … + 100
Rule
Lower limit
Upper limit
c) 4 + 7 + 10 + … + 46
Rule
Lower limit
Upper limit
1 1 1
d) 1+ + + + ...
8 27 64
Rule
Lower limit
Upper limit
Section 12.2
Arithmetic Sequences and Series
An arithmetic sequence is a sequence where difference between consecutive terms is _________.
We keep adding or subtracting the same number - the common difference, d.
For example: 1, 5, _____, _____. The common difference, d, is__________.
4, 2, _____, _____ . The common difference, d, is__________.
Ex. 1 Is the sequence arithmetic? If so, find the common difference and the next 2 terms.
1, 2, 4, 8, 16, …
13, 7, 1, -5, …
Arithmetic?_________
d ______
next 2 terms:___________
Arithmetic?_________
d ______
next 2 terms:___________
Arithmetic Sequence has an n th term given by
Where a1 is the first term, d is the common difference, n represents
the number of the term you are
an = a1 + (n -1)d
looking for and an is the value of
th
n term. Explicit rule appears linear.
Ex. 2 Write a rule for the n th term of each sequence. Then find seventh term.
a) 32, 47, 62, 77, …
b) 5 7 9 11 13
, , , , , ...
2 2 2 2 2
Ex. 3
Find the formula for the nth term of an arithmetic Two terms of arithmetic sequence are: a10 =148
sequence if the common difference is 5 and the
and a44 = 556 . Write rule for nth term. Find for
2nd term is 12.
what value of n an =124 .
Arithmetic Series
An Arithmetic Series is the expression formed by adding terms of a finite arithmetic sequence.
Ex. 4 Gauss was given this detention assignment when he was in 3rd grade.
“Add all of the integers from 1 to 100 and then you can go home.”
He was able to do it in just 30 seconds. Can you? Try it!
1+2+3+4+5+…+97+98+99+100 = ??????
The Sum of an Arithmetic Series is:
Where S n is the sum of first n terms of an arithmetic series, a1 is the
first term, an is the n th term, n is the number of terms, d is common
difference.
Ex. 5 Find the sum of the arithmetic sequence without adding up all of the terms!
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
Ex. 6 Find the sum of the 1st n terms of the arithmetic series.
2, 8, 14, 20, …
Ex. 7 Evaluate:
32
 2i  8
i 1
n = 25
5 7 9 11 13
, , , , , ...
2 2 2 2 2
n=32