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

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