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# Sequences and Series

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by

Tweet## Jennifer Lyon

on 4 March 2013#### Transcript of Sequences and Series

Sequences and Series Miss Lyon Pre-Calculus Sequences Series Arithmetic Series Geometric Series One Last Thing Means Geometric Sequences Formulas Arithmetic Sequences Sequence Explicit Formula Recursive Formula Review A sequence is a list of ordered numbers.

Each number in a sequence is called a term and we write them like .

Example: 1, 3, 5, 7, 9 ...

=7 A recursive formula defines a sequence by relating each term to the ones before it.

Ex: An explicit formula expresses the nth term using n.

Ex: In an arithmetic sequence, the difference between consecutive terms (terms in a row) is constant. In a geometric sequence the ratio between consecutive terms is constant. The arithmetic mean of any two numbers is the average

If you have 3 consecutive terms in an arithmetic sequence, the middle term will be the arithmetic mean of the first and third

Ex: An arithmetic series is one whose terms are an arithmetic sequence

To find the sum ( ) of a finite arithmetic series

( ) use

Ex: 6+9+12+15+18 A geometric series is one whose terms are a geometric sequence

To find the sum ( ) of a finite geometric series

( where r≠1) use

Ex: 3+6+12+24+48+96 We can write a series with the summation symbol

We need to use limits, the least and greatest values of n

Ex: Find the sum of the first 33 terms of 3+6+9+... The difference between consecutive terms is called the common difference and is often represented by d. Ex: 6, 12, 18, 24

12-6 = 6 18-12=6 24-18=6

This is an arithmetic sequence with d=6 Is 2, 4, 8, 16 an arithmetic sequence? The ratio is called the common ratio and represented by r. Ex: 5, 15, 45, 135

15/5=3 45/15=3 135/45=3

This is a geometric sequence with r=3 Is 15, 30, 45, 60 a geometric sequence? d=common difference r=common ratio Example: 3, 6, 9, 12 Arithmetic sequence, d = 3 Recursive:

Explicit: Example: 1, 7, 49, 343 Geometric sequence, r=7 Recursive:

Explicit Means The geometric mean of any two numbers is the square root of the products

If you have 3 consecutive terms in a geometric sequence, the middle term will be the geometric mean of the first and third

Ex: A series is the expression for the sum of the terms of a sequence

Finite sequences and series have a certain amount of terms that you can count. Infinite sequences and series continue without end

Ex: Finite Sequence: 6, 9, 12, 15 Finite Series: 6+9+12+15 Infinite Sequence: 3, 7, 11 ... Infinite Series: 3+7+11+... Infinite Geometric Series An infinite geometric series can either converge or diverge

The series converges, gets closer and closer, to a sum (S) when |r|<1

Ex:

The series diverges, or gets huge, when |r|>1

Ex: 4+8+16+...

r=2>1 so the series diverges

We can find the sum (S) of an infinite geometric series if it converges

Use the following formula:

Ex:

Full transcriptEach number in a sequence is called a term and we write them like .

Example: 1, 3, 5, 7, 9 ...

=7 A recursive formula defines a sequence by relating each term to the ones before it.

Ex: An explicit formula expresses the nth term using n.

Ex: In an arithmetic sequence, the difference between consecutive terms (terms in a row) is constant. In a geometric sequence the ratio between consecutive terms is constant. The arithmetic mean of any two numbers is the average

If you have 3 consecutive terms in an arithmetic sequence, the middle term will be the arithmetic mean of the first and third

Ex: An arithmetic series is one whose terms are an arithmetic sequence

To find the sum ( ) of a finite arithmetic series

( ) use

Ex: 6+9+12+15+18 A geometric series is one whose terms are a geometric sequence

To find the sum ( ) of a finite geometric series

( where r≠1) use

Ex: 3+6+12+24+48+96 We can write a series with the summation symbol

We need to use limits, the least and greatest values of n

Ex: Find the sum of the first 33 terms of 3+6+9+... The difference between consecutive terms is called the common difference and is often represented by d. Ex: 6, 12, 18, 24

12-6 = 6 18-12=6 24-18=6

This is an arithmetic sequence with d=6 Is 2, 4, 8, 16 an arithmetic sequence? The ratio is called the common ratio and represented by r. Ex: 5, 15, 45, 135

15/5=3 45/15=3 135/45=3

This is a geometric sequence with r=3 Is 15, 30, 45, 60 a geometric sequence? d=common difference r=common ratio Example: 3, 6, 9, 12 Arithmetic sequence, d = 3 Recursive:

Explicit: Example: 1, 7, 49, 343 Geometric sequence, r=7 Recursive:

Explicit Means The geometric mean of any two numbers is the square root of the products

If you have 3 consecutive terms in a geometric sequence, the middle term will be the geometric mean of the first and third

Ex: A series is the expression for the sum of the terms of a sequence

Finite sequences and series have a certain amount of terms that you can count. Infinite sequences and series continue without end

Ex: Finite Sequence: 6, 9, 12, 15 Finite Series: 6+9+12+15 Infinite Sequence: 3, 7, 11 ... Infinite Series: 3+7+11+... Infinite Geometric Series An infinite geometric series can either converge or diverge

The series converges, gets closer and closer, to a sum (S) when |r|<1

Ex:

The series diverges, or gets huge, when |r|>1

Ex: 4+8+16+...

r=2>1 so the series diverges

We can find the sum (S) of an infinite geometric series if it converges

Use the following formula:

Ex: