We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
125
1
avatar+829 

please explain, many thanks

 Nov 26, 2018
 #1
avatar+99523 
+3

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

5 Online Users