Pre-Calculus
11.1 Sequences and Summation Notation
Name __________________________________
Date ____________________ Period ________
Sequences:
Infinite Sequence – a function whose domain is the set of positive integers. The function values
a1 , a2 , a3, ..., an ,...
are the terms of the sequence.
Finite Sequence – a sequence whose domain consists of the first n positive integers only.
Recursive Sequence – a sequence where the first one or more terms are given. All other terms of the
sequence are then defined using previous terms.
In terms of a n , a n 1 indicates __________________ and a n 1 indicates _________________
Ex: Write the first 5 terms of the sequence given by:
1. an 
(1)n
2n  1
2. a1  3, ak 1  2(ak  1)
 n 

n  3
3. 
Factorial – If n is a positive integer, n factorial is defined by: n! n  (n  1)  (n  2)  ....  3  2  1
n!  n(n  1)!
NOTES:
0! = 1
- and -
Ex: Evaluate the following:
4. 5!
5.
5!
8!
Ex: The given pattern continues. Write down the nth term of each sequence suggested by its pattern.
6. 2, 5, 10, 17, ….
7. –1, 4, -9, 16, -25, 36, ….
Summation Notation –
The sum of the first n terms of a sequence is represented by
n
a
i
i 1
 a1  a2  a3  ....  an
where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of
summation. (note: lower limit does not have to be 1)
Ex: Find the sum of each sequence.
5
8.
 (2i  3)
i 1
Ex: Express the sum using sigma notation.
10.
1 2 6 24
720
  
 .... 
2 4 8 16
64
4
9.
 2i
i 0
2
Properties of Summation:
n
1.
c 
cn
k 1
2.
n
n
k 1
k 1
 c ak  c ak
n
3.
 ak  bk 
k 1
n
4.
 ak 
k 1
n
5.
k

k 1
n
6.
k
2

k 1
n
7.
k3 
k 1

n
n
 ak 
b
k 1
j
 ak 
k 1
k 1
n
a
k  j 1
k
where 0  j  n
k
n(n  1)
2
n(n  1)( 2n  1)
6
n 2 (n  1) 2
4
Find the sum of each sequence:
12
11.
 (3k )
k 1
10
12.
k
k 1
2
1
Pre-Calculus
11.2 Arithmetic Sequences
Name _________________________
Date ______________ Period ______
Arithmetic Sequences
A sequence is arithmetic if the difference between consecutive terms is the same. Thus, the sequence
is arithmetic if there is a number d such that:
a1 , a2 , a3 ,...., an ,....
a2  a1  d ,
a3  a2  d ,
a4  a3  d ,
and so on. The number d is the common difference of the arithmetic sequence.
Examples of arithmetic sequences: find the common difference, d.
13. 5, 10, 15, 20, 25, 30, 35, ….
14. 17, 14, 11, 8, 5, 2, -1, -4, ….
The nth term of an arithmetic sequence has the form:
an  a1  (n  1)d
or
an  dn  a 0
where d is the common difference between consecutive terms of the sequence and
a 0  a1  d
Ex: Write the first 5 terms of the arithmetic sequence, find the common difference, and write the nth term of the
sequence as a function of n.
15. a1  15, ak 1  ak  6
16. a3  94, a6  85
17. Find the sum of the even integers from 2 to 40.
There must
be an easier
way!
The Sum of a Finite Arithmetic Sequence
The sum of a finite arithmetic sequence with n terms is given by
Sn 
n
(a1  an )
2
Ex: Find each sum.
100
18. Find the nth partial sum of:
19.
 7n
n  51
-6, -2, 2, 6,….. if n = 40
Assignment: Pp. 798 – 799 # 3, 5, 13, 15, 21 – 29 odd, 35 – 41 odd, 51, 55
PRE-CALCULUS
11.3 Geometric Sequences
NAME _________________________
DATE _______________ PER _____
Geometric Sequence: sequence formed by multiplying the same number to each term to create the next term.
a1, a2, a3, a4, … is geometric if an = an-1(r) for all integers n > 1.
r = common ratio =
an
an 1
an = a1  rn-1
nth Term of a Geometric Sequence:
Geometric Means – numbers between 2 numbers in a geometric sequence!
20. Find 6 geometric means between 11 and 1408.
11, _____, _____, _____, _____, _____, _____, 1408
Note: “Mean Proportional” or “Geometric Mean” is a single mean between 2 #’s. The Geometric Mean
between 2 positive numbers is positive, the Geometric Mean between 2 negative numbers is negative!
Geometric Partial Sum:

21. Find S6 if

k 1
3
4k
Sn =

a1 1  r n
1 r

Infinite Geometric Series:

n
If r < 1, then r converges to zero.
EX]

n0

If r > 1, then rn diverges.
EX]
1
 
 3
 6
n
n
exists! (the sum is 0.5)
D.N.E. (Does Not Exist),
n0
because it increases without bound
Infinite Geometric Partial Sum for converging series
S 
a1
1 r

22. Find

2n
n 1

23. Find

k 1
 2
7 
 3
k 1
24. Rewrite the repeating decimal as a fraction in simplest form: 14.26 = 14.262626…
Home Work: Pp. 808 – 810 # 9, 11, 17, 19 – 29 odd, 41, 45, 47, 49, 53, 55, 59, 63, 65, 71, 73, 75, 77, 79