Sequences and Summations - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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1

Sequences and Summations

2

Definitions

• Sequence: an ordered list of elements

• Like a set, but:
  • Elements can be duplicated
  • Elements are ordered

4

Sequence examples

• a (^) n = 3 n
  • The terms in the sequence are a 1 , a 2 , a 3 , …
  • The sequence { an } is { 3, 6, 9, 12, … }
• b (^) n = 2

n

• The terms in the sequence are b 1 , b 2 , b 3 , …
• The sequence { bn } is { 2, 4, 8, 16, 32, … }
• Note that sequences are indexed from 1
• Not in all other textbooks, though!

5

Geometric vs. arithmetic

sequences

• The difference is in how they grow
• Arithmetic sequences increase by a constant amount
  • a (^) n = 3 n
  • The sequence { a (^) n } is { 3, 6, 9, 12, … }
  • Each number is 3 more than the last
  • Of the form: f ( x ) = dx + a
• Geometric sequences increase by a constant factor
  • b (^) n = 2 n
  • The sequence { b (^) n } is { 2, 4, 8, 16, 32, … }
  • Each number is twice the previous
  • Of the form: f ( x ) = ar x

7

Reproducing rabbits

• You have one pair of rabbits on an island

• The rabbits repeat the following:
  • Get pregnant one month
  • Give birth (to another pair) the next month
• This process repeats indefinitely (no deaths)
• Rabbits get pregnant the month they are born

• How many rabbits are there after 10

months?

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Reproducing rabbits

• First month: 1 pair
  • The original pair
• Second month: 1 pair
  • The original (and now pregnant) pair
• Third month: 2 pairs
  • The child pair (which is pregnant) and the parent pair (recovering)
• Fourth month: 3 pairs
  • “Grandchildren”: Children from the baby pair (now pregnant)
  • Child pair (recovering)
  • Parent pair (pregnant)
• Fifth month: 5 pairs
  • Both the grandchildren and the parents reproduced
  • 3 pairs are pregnant (child and the two new born rabbits)

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Reproducing rabbits

• Note the sequence:

• The Fibonacci sequence again

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Fibonacci sequence

• Another application:

• Fibonacci references from

http://en.wikipedia.org/wiki/Fibonacci_sequence

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Determining the sequence formula

• Given values in a sequence, how do you

determine the formula?

• Steps to consider:
  • Is it an arithmetic progression (each term a constant

amount from the last)?

• Is it a geometric progression (each term a factor of

the previous term)?

• Does the sequence it repeat (or cycle)?
• Does the sequence combine previous terms?
• Are there runs of the same value?

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Determining the sequence formula

a) 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, …

The sequence alternates 1’s and 0’s, increasing the number of 1’s and 0’s each time

b) 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, …

This sequence increases by one, but repeats all even numbers once

c) 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, …

The non-0 numbers are a geometric sequence (2 n ) interspersed with zeros

d) 3, 6, 12, 24, 48, 96, 192, …

Each term is twice the previous: geometric progression

a (^) n = 3*2 n-

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Useful sequences

• n

2

• n

3

• n

4

n

n

• n! = 1, 2, 6, 24, 120, 720, …

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• A summation:

or

• is like a for loop:

int sum = 0;

for ( int j = m; j <= n; j++ )

sum += a(j);

Summations

=

n

j m

a j ∑ = j

n j ma

index of summation

upper limit

lower limit

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Evaluating sequences

• Let S = { 1, 3, 5, 7 }
• What is Σj∈ S j

• What is Σj∈ S j

2

2

• 3

2

• 5

2

• 7

2 = 84

• What is Σj∈ S (1/ j )

• What is Σj∈ S 1

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• Sum of a geometric

series:

• Example:

10 10 1

0

=

∑ j

n

Summation of a geometric series

=

∑

( 1 ) if 1

if 1 1

1

0 n a r

r r

ar a

ar

n n

j

j