Sequences and SeriesSequences and Series(T) Students will know the form of an Arithmetic (T) Students will know the form of an Arithmetic

sequence. sequence. Arithmetic Sequence: There exists a common Arithmetic Sequence: There exists a common

difference (d) between each term.difference (d) between each term. Ex: 2, 6, 10, 14,…Ex: 2, 6, 10, 14,… d = ?d = ? Ex: 17, 10, 3, -4, -11,…Ex: 17, 10, 3, -4, -11,… d = ?d = ? Ex: a, a+d, a+2d, a+3d, a+4d,…Ex: a, a+d, a+2d, a+3d, a+4d,…

General term: General term: 1 ( 1)nt t n d

(T) Students will know the form of a geometric (T) Students will know the form of a geometric sequence.sequence.

Geometric Sequence: There exists a common Geometric Sequence: There exists a common ratio (r) between each term.ratio (r) between each term. Ex: 1, 3, 9, 27, 81 r = ?Ex: 1, 3, 9, 27, 81 r = ? Ex: 64, -32, 16, -8, 4 r = ?Ex: 64, -32, 16, -8, 4 r = ? Ex: a, ar, ar^2, ar^3, ar^4,…Ex: a, ar, ar^2, ar^3, ar^4,…

General term: General term: ( 1)

1n

nt t r

(T) Students know the recursive definition.(T) Students know the recursive definition. Recursive definitions: The next value of the Recursive definitions: The next value of the

sequence is determined using the previous sequence is determined using the previous term.term. Ex: Ex:

Explicit definitions were given in the previous section.Explicit definitions were given in the previous section.

1

1

2

3

3

2 1

2(3) 1 7

2(7) 1 15

n n

t

t t

t

t

Ex: 23, 20, 17, 14,…Ex: 23, 20, 17, 14,…

Recursive def: Recursive def:

Explicit def:Explicit def:

1

1

23

3n n

t

t t

26 3nt n

(T) Students understand the concept of a series.(T) Students understand the concept of a series.

Arithmetic and Geometric Series: A series is the Arithmetic and Geometric Series: A series is the sum of the terms of a sequence.sum of the terms of a sequence.

Finite sum of an arithmetic series:Finite sum of an arithmetic series:

Illustrate proof:Illustrate proof:

1( )

2n

n

n t tS

(T) Students can find the sum of a series.(T) Students can find the sum of a series.

Sum of a finite geometric series:Sum of a finite geometric series:

Where r can not equal 1.Where r can not equal 1.

Illustrate proof.Illustrate proof.

1(1 )

1

n

n

t rS

r

(T) Students understand the concept of limits.(T) Students understand the concept of limits. Infinite Sequences: A sequence that continues Infinite Sequences: A sequence that continues

forever.forever.

Ex: ½, ¼, 1/8, 1/16, …, (1/2)^n,…Ex: ½, ¼, 1/8, 1/16, …, (1/2)^n,…

Limits: The sequence approaches some Limits: The sequence approaches some number but never reaches it. On a graph it is an number but never reaches it. On a graph it is an asymptote.asymptote.

Ex: Ex:

10

2

( 1)1 1

lim

lim

n

n

n

n n

Theorem: Theorem:

(T) Students know the formula for an infinite (T) Students know the formula for an infinite geom. Series.geom. Series.

Sum of an infinite geometric series:Sum of an infinite geometric series:

n

If r 1, then 0lim nr

1

1

If |r| < 1, the infinite geometric series converges

tto the sum S=

1-rIf |r| 1, and t 0, then the series diverges.

(T) Students can use mathematical induction for (T) Students can use mathematical induction for proofs.proofs.

Mathematical Induction: Let S be a statement in Mathematical Induction: Let S be a statement in terms of a positive integer n.terms of a positive integer n. Show that S is true for n=1Show that S is true for n=1 Assume that S is true for n=k, where is a positive Assume that S is true for n=k, where is a positive

integer, and then prove that S must be true for n = k + 1.integer, and then prove that S must be true for n = k + 1.

Prove that Prove that

Prove that n^3 + 2n is a multiple of 3.Prove that n^3 + 2n is a multiple of 3.

1 1 1 1...

1 2 2 3 3 4 ( 1) 1

n

n n n