(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that 3 divides n^{3}+ 2n whenever n is a positive integer.

2. Relevant equations

3. The attempt at a solution

Basis Step :

P(1) : [1^{3}+ 2(1) ] /3

[1+2] /3

[3]/3

1

Since 3/3 =1, P(1) is true

Inductive Step:

[ k^{3}+ 2k ] /3 = [(k +1)^{3}+ 2(k+1) ] /3

[k(k^{2}+ 2k)] /3 = [(k)^{3}+ (3k)^{2}+ 5k +3] /3

1. The problem statement, all variables and given/known data

Prove that f_{1}^{2}+ f_{2}^{2}+...+ f_{n}^{2}when n is a positive integer.

2. Relevant equations

The Fibonacci numbers f_{0}, f_{1}, f_{2}..., are defined by the equations f_{0}= 0, f_{1}= 1 and f_{n}= f_{n-1}+ f_{n-2}for n = 2,3,4...

3. The attempt at a solution

Basic Step:

P(1) : f_{1}^{2}= f_{1}* f_{2}

1^{2}= (1)*(1)

1= 1

Since f_{1}^{2}= f_{1}* f_{2}, P(1) is true.

Inductive Step:

f_{k}^{2}= f_{k}* f_{k+1}

f_{k}^{2}+ f_{k+1 }^{2}

= f_{1}+f_{k+1}+ (f_{k+1})

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# Homework Help: Mathenatucak Induction Problems in discrete math

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