Math Induction: Verifying Sequence Properties with Induction and Notation - Prof. Jandelyn, Study notes of Discrete Structures and Graph Theory

This chapter explores mathematical induction, a powerful technique used to verify properties of sequences. Learn about infinite and finite sequences, finding explicit formulas, summation and product notation, nesting, telescoping series, merging and splitting, distribution properties, and factorials.

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Pre 2010

Uploaded on 07/29/2009

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Chapter 4
Mathematical Induction
Used to verify a property of a sequence
2,4,6,8,… for i >= 1 a
i
= 2i
infinite sequence with infinite distinct values
for i >= 1 b
i
= (-1)
i
infinite sequence with finite distinct values
for 1<=i<=6 c
i
= i+5
finite sequence (with finite distinct values)
pf3
pf4
pf5
pf8

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Chapter 4

Mathematical Induction

  • Used to verify a property of a sequence• 2,4,6,8,…

for i >= 1

a

i^

= 2i

  • infinite sequence with infinite distinct values
    • for i >= 1

b

i^

i

  • infinite sequence with finite distinct values
    • for

1<=i<=

c

i^

= i+

  • finite sequence (with finite distinct values)

Finding the Explicit Formula

  • Figure the formula of this sequence• Different sequences with same initial values

,...

(^125) ,

16

1

, 1 9 , 1 4

, 1

2

) 1

(

1

(^02)

=

=

=

k

k

b

k

k a

k k

Variable ending point

  • n as the index of the final term• for n = 2• for n = 3



n

k^

k

k n

1

Nesting of Sum/Product Notation• Variations (same or different??):

 ^ =

=

J j

n i

ij j

Y

1

2

1

)

(

 ^ =

J j

n i

ij

j

Y

)

(

 

J j
n i
ij

j

Y

Properties

  • Merging and Splitting• Distribution







=

n^ m k

k

k

n^ m k

k

n^ m k

k^

b

a

b

a

)

(

)

(

*^

k k

n^ m k

k

n^ m k k

n^ m k

b a

b

a

Π

Π

Π

=





=

n^ m

k

k

n

m

k

k^

a

c

a

c

)

(







=

n^ i k

k

i^ m k

k

n^ m k

k^

a

a

a

k

n^ i k k

i^ m k

k

n^ m k

a

a

a

Π

Π

Π

=

Factorial

  • n! = n(n-1)(n-2)2*1• Definition

0! = 1n! = n*(n-1)!