Download Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Name

Algebra 2 AII.2 Sequences and Series Notes
Mrs. Grieser
Name: _____________________________________ Date: _______________ Block: _______
Sequences and Series

Definitions:
sequence
an ordered list of numbers
term
each number in a sequence
series
the sum of a sequence
Arithmetic Sequences

Domain of a sequence: natural numbers {1, 2, 3, …}

Range: value of the terms in the sequence

Some sequences show patterns; some do not

Finite sequences contain a finite (countable) number of terms

Infinite sequences contain an infinite (uncountable) number of terms

Arithmetic sequences contain a pattern where a fixed amount is added from one term
to the next (common difference d) after the first term

Geometric sequences contain a pattern where a fixed amount is multiplied from one
term to the next (common ratio r) after the first term

Arithmetic sequence examples:
o 1, 4, 7, 10, 13, 16, …
o Domain: _______________________
o Range: ________________________
o Graph shown at right
o common difference d = _________
o The graph of an arithmetic sequence is
______________
o Find the common difference (d) for the following
arithmetic sequences:

a) 15, 10, 5, 0, …
d = _______

b) 1,

Write the next 2 terms of each sequence: a) ______________ b) _____________

Write the 8th term of each sequence:
1
1
, 0 , - , …. d = _______
2
2
a) ______________ b) _____________
Algebra 2 AII.2 Sequences and Series Notes

Mrs. Grieser Page 2
Notation:
o an = nth term of an arithmetic sequence
o a1, a2, …, an : terms of an arithmetic sequence

Finding Terms in a Sequence
o Find the 8th term in the sequence: 5, 9, 13, 17, …

d = _____________

a8 = ____________
o Is there a pattern?

a1 = 5

a2 = a1 + d

a3 = a2 + d = a1 + d + d = ___________

a4 = a3 + d = a2 + d + d = a1 + d + d + d = ___________

an = _____________
To find the nth term in an arithmetic sequence:
an = a1 + (n – 1)d
where a1 is the first term of the sequence,
d is the common difference, n is the number of the term to find
o You try…Find the requested term in the sequence:
a) Find the 7th term:
3, 9, 15, 21, …

b) Find the 10th term:
3, 5, 7, 9, …
c) Find a7:
a1 = 3x and d = -x
d) If a4 = 6 and d = -2,
find a5
Sequence Formulas
o Rather than writing out terms, we often use a formula or rule for a sequence
o Example: an = 2n
o Write out the first 6 terms: ___________________________________
o What is a10? ___________
o Example: Given an = 6n + 3

Find the first 5 terms _______________________________

What is the common difference d? ________________

What is the 10th term of the sequence? ____________

What is a15? ___________
Algebra 2 AII.2 Sequences and Series Notes
Mrs. Grieser Page 3
o Example: Find a formula for the sequence 1, 3, 5, 7, …

General formula for the nth term: _____________

What is the common difference? ________

Substitute common difference for d, and simplify _________________
o Example: One term of an arithmetic sequence is a19 = 48. The common difference is
d = 3.

Use the general rule for the nth term: _____________________
o Example: Insert three arithmetic means between 7 and 23

An arithmetic mean is the term between any two terms of an arithmetic
sequence. It is simply the average (mean) of the given term.

Use the general formula for the nth term to find d, the common difference:

Now use d to find 3 arithmetic means: ________________________
o You Try…Answer each question, then find a20 for each sequence.
a) Find the first 6 terms of
the sequence:
an = 6 - n
b) Write a rule for the
sequence given a11=-57
and d = -7
c) Write a rule for the
sequence that has a7=26
and a16=71
Arithmetic Series

An arithmetic series is the sum of an arithmetic sequence: Sn=
n
a
i 1

Summation Notation Review
o Read: the sum as i goes from 1 to infinity of ai
o Can be finite or infinite
o index starts at lower limit of summation, then
increments by 1 until upper limit is reached
o Index doesn’t have to be i; can be any letter
o Index doesn’t have to start at 1
i
Algebra 2 AII.2 Sequences and Series Notes
Mrs. Grieser Page 4
o Examples:

4
 2i
a)
 2i
i 1
b)
i 1
5
 (m
2
 1)
c)
m 1
o Sum of a constant:
n
k
6
4
1
d)
i
5
e)
i 1
j 1
k
2
k 1
= _____________
i 1
 Find
12
5
______________
i 1
o Sum of the first n numbers:
n
i
= __________
i 1
 Find
50
k
______________
k 1
o Sum of the squares of the first n numbers:
n
i
2
= __________
i 1
 Find
18
k
2
______________
k 1
o Other properties of summation:

n
n
i 1
i 1
 kai  k  ai , where k is a constant
10
 4i
Find

i 1

n
n
n
 ( a  b )  a   b
i
i 1

Find
i
i 1
i
i 1
i
8
 (i  7)
i 1
o You Try…Find the values of the summations:
a)
7
 8i
i 1
b)
12
 (k  10)
k 1
10
c)
 (k 2  1)
k 1
20
d)
( j
j 1
2
 j  5)
Algebra 2 AII.2 Sequences and Series Notes

Mrs. Grieser Page 5
Sums of a Finite Arithmetic Series
o The sum of the first n terms of an arithmetic series is n times the mean of the first
and last terms:
a a 
Sn = n 1 n 
 2 
 Find
10
 (3i  1)
i 1

n = _____, a1 = _______, an = _______

Use formula:
 You try: Find
20
 (3  5i)
i 1

n = _____, a1 = _______, an = _______

Use formula:
o What if you don’t know an, but you do know the common difference, d?
 a  an 
 Sn = n 1

 2 
 an = a1 + (n-1)d
 a  an 
 a  a1  (n  1)d 
 2a  (n  1)d 
 Substitute: Sn = n 1
 = n 1
  n 1

2
2




 2 
o Example: Find the sum of the first 15 terms of the sequence 3, 7, 11, …
 n = _____________
a1 = _______________ d = _______________
 Use formula:
 Sum: ___________
o You Try…
a) Find
12
 (2  7k )
b) Find the sum of the first 10 terms of the sequence: 9, 5, 1, …
k 1
c) During a high school spirit week, students dress up in costumes, with a cash
prize being given each day for the best costume. The organizing committee has
$1,000 to give away over 5 days. The committee wants to increase the amount
by $50 each day. How much should the committee give away on the first day?