proof by induction, Schemes and Mind Maps of Calculus

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PROOF

BY

INDUCTION

SUMMATION

RESULTS

Question 3 (+)**

Prove by induction that

1

n

r

r r n n n

=



, n ≥ 1 , n ∈
.

proof

Question 4 (+)**

Prove by induction that

2

2

n

r

r r n n n n

=



, n ≥ 2 , n ∈
.

FP1-E , proof

Question 7 ()*

Prove by induction that

2

1

n

r

r r n n

=



, n ≥ 1 , n ∈
.

proof

Question 8 ()*

Prove by induction that

1

n

r

n

r r n

=



, n ≥ 1 , n ∈
.

proof

Question 9 ()*

Prove by induction that

( )

1

1

n n

r

r

−

=



, n ≥ 1 , n ∈
.

proof

Question 11 (+)*

Prove by induction that

2

1

n

r

n

r^ n

=



, n ≥ 1 , n ∈
.

FP1-N , proof

Question 12 (+)*

Prove by induction that

1

1

n

r n

r

r n

=

× = + −



, n ≥ 1 , n ∈
.

proof

Question 13 (+)*

Prove by induction that

1

n

r n

r

r n

=

+ × = ×

  

, n ≥ 1 , n ∈
.

proof

Question 14 (+)*

If n ≥ 1 , n ∈
, prove by induction that

1 1!× + 2 × 2! + 3 × 3! + ... + n × n! = ( n +1! ) − 1.

proof

Question 16 (****)

Prove by induction that

2 2

2 2 2

1

n

r

r n

r r n

=



, n ≥ 1 , n ∈
.

FP1-R , proof

Question 17 (****)

Prove by induction that

1

n

r

r r n n n

=



, n ≥ 1 , n ∈
.

proof

Question 20 (****)

Prove by mathematical induction that if n is a positive integer then

( ) (^) ( )

2 2

1

n

r

r n n n

=



You may not use other methods of proof in this question.

FP1-Y , proof

Question 21 (****)

Prove by mathematical induction that if n is a positive integer then

1

n

r

r n n

r r r n n

=

 (^) + + + +

You may not use other methods of proof in this question.

FP1-L , proof

DIVISIBILITY

RESULTS

Question 1 ()**

n f n = + , n ∈
.

Prove by induction that f ( n ) is divisible by 6 , for all n ∈
.

proof

Question 2 ()**

n f n = + , n ∈
.

Prove by induction that f ( n ) is divisible by 5 , for all n ∈
.

proof