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Inductive Proofs for Various Mathematical Sequences, Study notes of Algorithms and Programming

Eastern Washington University (EWU)Algorithms and Programming

Inductive proofs for various mathematical sequences, including the sum of integers, odd numbers, even numbers, fibonacci numbers, power series, sum of cubes, and sum of squares.

Typology: Study notes

Pre 2010

Uploaded on 08/17/2009

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bg1
Specimen Inductive Proofs
by Tim Rolfe
Sum of integers:
QED
[simplify] )1(
2
1
/2]out [factor )21(
2
1]by mu ltiply ;hypothesis [Inductive
2
2
)1(
2
1
e][Recurrenc )1()(
[Basis]010
2
1
0)0(
)1(
2
1
)( :Prove
)1(:0
0:0
)(
11
1
1
1
1
1
nn
nn
n
nnn
nnSnS
S
nnnS
nnSn
n
jnS
n
j
Sum of odd numbers:
QED
[Simplify]
quadratic] the[Expand 1212
]hypothesis [In ductive )12()1(
e][Recurrenc )12()1()(
000)0(
)( :Prove
)12()1(:0
0:0
12)(
2
2
2
2
2
1
n
nnn
nn
nnSnS
S
nnS
nnSn
n
jnS
OO
O
O
n
jO
O
Sum of even numbers:
)1()1(
2
1
2
aboveproven been has which )(222)(
result)earlier (using proof eAlternativ
QED
[Simplify] )1(
]out [Factor )21(
]hypothesis [Inductive 2)1(
e][Recur renc 2)1()(
0100)0(
)1()( :Prove
2)1(:0
0:0
2)(
1
11
1
nnnn
nSjjnS
nn
nnn
nnn
nnSnS
S
nnnS
nnSn
n
jnS
n
j
n
j
E
EE
E
E
n
jE
E
Sum of Fibonacci numbers:
QED] :)2( ofn [Definitio1)2(
vity][Associati1)()1(
]hypothesis [Inductive)(1)1(
e][Recurrenc)()1()(
0111)2()0(
1)2()( Prove
)()1(:0
0:0
)()(
0
nFibnFib
nFibnFib
nFibnFib
nFibnSnS
SS
nFibnS
nFibnSn
n
jFibnS
FibFib
FibFib
Fib
Fib
n
j
Fib
Power Series:
QED
] eliminate[
1
1
multiply] denom.; [common
1
)1()(
rearrange] 1;by multiply [
1
1
1
1
]hypothesis inductive[
1
1
e][recurrenc)1()(
1
1
1
1)0(
1
1
)( :P rove
)1(:0
1:0
)(
1
1
1
1
0
nn
n
nnn
n
n
n
n
n
n
n
n
j
j
cc
c
c
c
ccc
c
c
c
c
c
c
c
c
cnPnP
c
c
P
c
c
nP
cnPn
n
cnP
Sum of cubes:
QED
]expression quadratic[Factor )1(
4
1
[Simplify] )12(
4
])1( expand /4;out [Factor 4)12(
4
1]by Multiply [
4
4
)1(
4
1
]hypothesis Inductive[)1(
4
1
e][Recurrenc )1()(
[Basis]010
4
1
0)0(
)1(
4
1
)( :Prove
)1(:0
0:0
)(
22
2
2
22
2
3222
322
3
33
22
3
22
3
3
1
3
3
nn
nn
n
nnnnn
n
nnnn
nnn
nnSnS
S
nnnS
nnSn
n
jnS
n
j
Sum of squares:
QED
]expression quadratic[Factor )12()1(
6
1
[Simplify] )132(
6
ms]linear ter two[Multiply 6)132(
6
/6]out [Factor )6)12()1((
6
1]by multiply Simplify;[
6
6
)122()1(
6
1
]h ypothesis Inductive[)122()11()1(
6
1
e][Recurrenc )1()(
[Basis]0110
6
1
0)0(
)12()1(
6
1
)( :Prove
)1(:0
0:0
)(
2
2
2
2
2
22
2
2
2
2
1
2
2
nnn
nn
n
nnn
n
nnnn
n
nnnn
nnnn
nnSnS
S
nnnnS
nnSn
n
jnS
n
j

Partial preview of the text

Download Inductive Proofs for Various Mathematical Sequences and more Study notes Algorithms and Programming in PDF only on Docsity!

Specimen Inductive Proofs

by Tim Rolfe

Sum of integers:

QED

( 1 ) [simplify]

2

( 1 2 )[factorout /2]

2

[Inductivehypothesis;multiplyby 1]

2

() ( 1 ) [Recurrenc e]

01 0 [Basis] 2

Prove: ( )

1 1

1

1

1 1

1

n n

n n

n

n n n

S n Sn n

S

Sn n n

n Sn n

n

S n j

n

j

Sum of odd numbers:

QED

[Simplify]

2 1 2 1 [Expandthe quadratic]

( 1 ) ( 2 1 ) [Inductivehypothesis ]

( ) ( 1 ) ( 2 1 ) [Recurrenc e]

Prove: ( )

n

n n n

n n

S n S n n

S

S n n

n S n n

n

S n j

O O

O

O

n

j O

O

Sum of even numbers:

() 2 2 2 () whichhasbeenproven above

Alternativeproof(usingearlier result)

QED

( 1 ) [Simplify]

( 1 2 ) [Factorout ]

( 1 ) 2 [Inductivehypothesis ]

() ( 1 ) 2 [Recurrenc e]

Prove: () ( 1 )

1 1 1

1

 

n n n n

S n j j S n

n n

n n n

n n n

S n S n n

S

S n n n

n S n n

n

S n j

n

j

n

j

E

E E

E

E

n

j E

E

Sum of Fibonacci numbers:

( 2 ) 1 [Definitionof ( 2 ): QED]

( 1 ) () 1 [Associati vity]

( 1 ) 1 () [Inductivehypothesis ]

() ( 1 ) () [Recurrenc e]

( 0 ) ( 2 ) 1 1 1 0

Prove () ( 2 ) 1

0 : ( 1 ) ( )

0 : 0

( ) ( )

   

   

   

  

   

  

  

  

Fibn Fib n

Fibn Fib n

Fibn Fib n

S n S n Fib n

S S

S n Fib n

n S n Fib n

n

S n Fib j

Fib Fib

Fib Fib

Fib

Fib

n

j

Fib

Power Series:

QED

[eliminate ]

1

[commondenom.; multiply]

1

[multiplyby1; rearrange] 1

[inductivehypothesis ]

1

() ( 1 ) [recurrenc e]

Prove: ( )

1

1

1

1

0

n n

n

n n n

n n

n

n

n

n

n

n

j

j

c c

c

c

c

c c c

c

c c c

c

c

c

c

Pn Pn c

c

c P

c

c

P n

n Pn c

n

Pn c

Sum of cubes:

QED

( 1 ) [Factorquadraticexpression ] 4

( 2 1 ) [Simplify] 4

( 2 1 ) 4 [Factorout /4;expand( 1 ) ] 4

[Multiplyby 1] 4

( 1 ) [Inductivehypothesis ]

4

() ( 1 ) [Recurrenc e]

0 1 0 [Basis] 4

Prove: ( )

2 2

2

2

2 2

2

2 2 2 3

2 2 3

3 3 3

2 2 3

2 2

3

1 3

3 3

n n

n n

n

n n n n n

n

n n n n

n n n

Sn Sn n

S

Sn n n

n S n n

n Sn j

n

j

Sum of squares:

QED

( 1 )( 2 1 ) [Factorquadraticexpression ] 6

( 2 3 1 ) [Simplify] 6

( 2 3 1 ) 6 [Multiplytwolinear ter ms]

6

(( 1 )( 2 1 ) 6 )[Factorout /6] 6

[Simplify;multiplyby 1] 6

( 1 )( 1 1 )( 2 2 1 ) [Inductivehypothesis ]

6

() ( 1 ) [Recurrenc e]

011 0 [Basis] 6

Prove: ( )

2

2

2

2

2 2 2

2

2

2 1 2

2 2

n n n

n n

n

n n n

n

n n n n

n

n n n n

n n n n

S n Sn n

S

S n n n n

n Sn n

n S n j

n

j