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