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Sequences and Series
Transcript of Sequences and Series
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
The series diverges, or gets huge, when |r|>1
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: