Discrete Structures Chapter 2 Part A Sequences

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Discrete StructuresChapter 2 Part ASequences

• Nurul Amelina Nasharuddin
• Multimedia Department

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Sequences

• Sequence is a set of (usually infinite number of)
  ordered elements a1, a2, , an,
• Eg 2, 4, 6, 8,
• Each individual element ak is called a term,
  where k is called an index
• The example above denotes an infinite sequence
• Sequences can be computed using an explicit
  formula ak k (k 1) for k gt 1
• a2 2 (2 1) 6, when k 2
• a3 3 (3 1) 12, when k 3
• a4 4 (4 1) 20, when k 4

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Sequences

• Finding an explicit formula given initial terms
  of the sequence 1, -1/4, 1/9, -1/16, 1/25,
  -1/36,
• Ans ak (-1)k1/ k2
• Sequence is (most often) represented in a
  computer program as a single-dimensional array

a1 a2 a3 a4 a5 a6
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Summation Operations

• Summation from k equals m to n, of ak where m ?
  n
• Computing summation Let a1 -2, a2 -1, a3 0

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Summation Operations

• Changing from summation notation to expanded
  form
• Changing from expanded form to summation
  notation

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Summation Operations

• Evaluating a1, a2, a3, , an for small n
• n1? 1/(1.2) 1/2
• n2? 1/(1.2) 1/(2.3) 2/3
• n3? 1/(1.2) 1/(2.3) 1/(3.4) 3/4
• Recursive definition If m and n are any integers
  with m lt n, then
• and

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Summation Operations

• Separating off the final term
• Adding on the final term
• Telescoping sum When writing sums in expanded
  form, you sometimes see all the terms cancel
  except for the first and last one.

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Product Operations

• Product from k equals m to n of ak
• Recursive definition If m and n are any integers
  with m lt n, then
• and

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Factorial Notation

• n factorial n! defined as the product of all
  integers from 1 to n,
• n! n ? (n 1) ? ? 3 ? 2 ? 1
• Zero factorial 0! 1
• Simplify the factorials

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Properties

• If am, am1, am2, and bm, bm1, bm2, are
  sequence of real numbers and c is any real
  number, then the following equations hold for any
  integer n ? m

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Change of Variable

• Observe that
• Transform a sum by changing variable
• Calculate new lower and upper limits
• When k 0, j k 1 0 1 1.
• When k 6, j k 1 6 1 7.
• The new sum goes from j 1 to 7

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Change of Variable

• Calculate new general term
• Since j k 1, then k j 1.
• Hence
• Finally put the steps together

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Exercise

• Compute
• Transform by making the change of variable
  j i 1
• Send in the answers on the next class!