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Section 11.1
Sequences
Sequence – list of values following a pattern
Arithmetic – from term to term there is a common difference we’ll call d
Geometric – from term to term there is a common ratio we’ll call r
Everything else
Definition (more mathematical)A sequence is a function whose domain is
the set of positive integers.A sequence is a function (input then output),
and it will have a graph. The positive integers are evaluated within
the function to give us the terms of the sequence.
Example: {an} = {(n-1)/n}
Identify the first 6 terms of
the sequence {an} = {(n-1)/n}
Calculator: seq(expression, variable, start, stop, increment) sequence located in LIST OPS 5
Identify the first 6 terms of the sequence {bn} = {(-1)n-
1(2/n)}
Calculator option 2: SEQ mode, “Y=“, nMin = 1 u(n)=expression, u(nMin)=2Look at the table
FactorialsThe factorial symbol, n!, is defined as
follows:
0! = 1 1! = 1
If n ≥ 2 is an integern! = n(n-1)(n-2) . . . (3)(2)(1) n! = n(n-
1)!
MATH PRB 4
ExamplesFind:
a) 5!
b) 10!
c) (4!)(6!)
RECURSIVE FORMULASWhen the sequence is defined by the
term(s) preceding the nth term
Must be given one or more of the first few terms
All other terms are then defined using the previous terms
MOST FAMOUS Fibonacci Sequencea0 = 1, a1 = 1, a2 = 2, a3 = 3, a4 = 5,…, ak =
ak-2 + ak-1
Summation NotationRather than write: a1 + a2 + a3 + . . . + an
we express the sum using summation notation.n
∑ak k=1
n
∑ak = a1 + a2 + a3 + . . . + ank=1
The index tells you where to start and end (bottom to top), although we often use k, it doesn’t matter
Rewrite the following
5
A) ∑ k-1
k=1
4
B) ∑ k! k=1
Write using summation notationa) 1 + 4 + 9 + 16 + . . . + 81
b) 1 + ½ + ¼ + 1/8 + . . . 1/(2n-1)
Properties of SequencesIf {an} and {bn} are 2 sequences and c is a
real number, then:
1.
2.
3.
4
5
6
7
8
Find the sum of each sequence
5
A) ∑ 3kk=1
4
B) ∑ k2 – 7k + 2 k=1
Things to watch for…(-1)n or (-1)n±1 when the sign changes each
term(2n) and (2n ± 1) for even and oddIf the terms differ by the same amount,
think linearIf the 2nd level terms differ by one amount,
think quadratic
ApplicationsAnnuity Formula
A0 = M (initial amount deposited)r = interest rate expressed as a percent in
decimal formN: number of compound periods per yearP: periodic deposit made at each payment periodAn = amount after n deposits have been made
1 1
1(1 )
n n n
n
rA A A P
NrA P
N
Applications
Amortization Formula
A0 = B (initial amount borrowed)r = interest rate expressed as a percent in
decimal formN: number of compound periods per yearP: periodic deposit made at each payment
periodAn = amount after n payments have been
made
1 1
1
12
(1 )12
n n n
n
rA A A P
rA P