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## Homework Statement

Prove that 3 divides n

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

## Homework Equations

## 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

## Homework Statement

Prove that f

_{1}

^{2}+ f

_{2}

^{2}+...+ f

_{n}

^{2}when n is a positive integer.

## Homework 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...

## 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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