Pre-Calculus Honors
8.1 Sequences & Summation Notation Notes
Mrs. Iverson
Name ___________________________
Date _______________ Period ______
Warm up: ACT Practice
What is the area, in square inches of the isosceles trapezoid below?
A.
B.
C.
D.
E.
15
20
27
36
45
Essential Question: How can I use this in real life?
Learning Targets: 4.8.1: Use sequence notation to write the terms of sequence.
4.8.2: Use factorial notation.
4.8.3: Use summation notation to write sums.
4.8.4: Find sums of infinite series.
4.8.5: Use sequences and series to model and solve real-life problems.
An infinite sequence is a function whose domain is the set of all positive integers. The function values
a1 , a2 , a3 ,........., an are terms of the sequence.
A finite sequence is a sequence just like a infinite sequence only it stops after n positive integers.
Definition of factorial: If n is a positive integer, n factorial is defined as n ! 1 2 3 4 (n 1) n .
A special case 0! 1
Example 1:
Write the first three terms of the sequence. (Find a1 , a2 , a3 )
an 4n 3
a)
Example 2:
a) an (1)
a16
b)
an
3n
4n
c)
Find the indicated value of the sequence
n 1
n(n 1)
n2
b) an
2n 1
a5
an
n!
n
Example 3: Use a graphing calculator to find the first five terms of the sequence.
b) an
a) an 2n(n 1)(n 2)
4n 2
n2
nth term of the sequence. Assume n begins with 1.
1 1 1 1
c)
2, 5, 10, 17…….
1, , , , ,........
4 9 16 25
Example 4: Write an expression for the most apparent
a)
3, 7, 11, 15, 19, …….
b)
A recursive sequence all terms of the sequence are defined using previous terms. You will need to be given
one or more of the first few terms to find a recursive sequence. A famous recursive sequence is the
Fibonnaci Sequence.
a0 1, a1 1, ak ak 2 ak 1
The Fibonacci Sequence:
Example 5:
Write the first five terms of the sequence defined recursively.
a1 15, ak 1 ak 3
a)
Example 6:
a1 3, ak 1 2(ak 1)
Simplify the ratio of factorials.
4!
7!
a)
b)
b)
25!
23!
c)
(n 2)!
n!
The Definition of Summation Notation: The sum of the first n terms of a sequence represented by
n
a
i 1
i
a1 a2 a3 ...... an where i is called the index of summation, n is the upper limit, and 1 is the lower
limit of summation.
Example 7:
Find the sum
5
6
a)
(3i 1)
i 1
b)
5
6
k 1
(k 1)(k 3)
c)
k 2
Example 8: Use the graphing calculator to find the sum.
3
x 1 x 1
Example 9:
A)
(1) k
k!
k 0
4
10
a)
b)
Use sigma notation to write the sum
5
5
5
5
...
11 1 2 1 3
1 15
B)
1 1 1
1
1 .....
2 4 8
128
Practice: p. 563-564 #1-11odd, 21,27,39,53,56,67,69,71,87,89,93,95
0
