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please explain, many thanks

 Nov 26, 2018
 #1
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Show that it is true for n = 1

 

(1)^3  - 1   =   0         and this is divisible by 3

 

Assume that it is true for    n = k

That is     k^3 - k     is divisible by 3

Note that we can write this as   k (k^2 - 1)  = k (k + 1) (k - 1) = (k - 1)(k)(k + 1)

 

Prove that it is true for    k + 1

 

(k + 1)^3  -   (k + 1)       factor

 

(k + 1) [ (k + 1)^2 - 1]

 

( k + 1) ( k^2 + 2k + 1  - 1 ]

 

(k + 1) [ k^2 + 2k ]

 

(k + 1) [ k ( k + 2) ] =

 

k (k + 1) (k + 2) =

 

k(k + 1) ( k + 3 - 1) =

 

k(k + 1) ( k - 1 + 3)  =

 

k(k + 1) (k - 1) + 3k(k+1)

 

(k-1) (k) (k + 1) + 3k(k + 1)

 

And we assumed that the first term was divisible by 3

And the second term is a multiple of 3, so it is also divisible by 3

 

So.... (k + 1)^3 - (k + 1) is divisible by 3

 

 

cool cool cool

 Nov 26, 2018

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