Use mathematical induction to prove each statement is true for all positive integers n:

5^(n)-1 is divisible by 4

n^(2)-n is divisible by 2

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n|d means "n divides d"



n=0:  ,  4 | 0

n=1:   ,  4 | 4



Assume true for n=k: i.e.  |    (hypothesis)



Let n=k+1:  

 

=  

= 

= 



4 |   (by the hypothesis) so  |  as well.



Therefore,  |  (if 4|P then 4|(P+4)) 

and the proof is complete.





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The other proof does not require induction  is always even, which is divisible by 2.   Follow the steps I did in first problem if you must have a proof by induction (show true for base case, assume hypothesis (n=k), and then show it leads to truth of the step case where n=k+1)







 

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