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Help! cannot solve P4 problem

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tantrik:
Hi friends,

I got in to this problem from Jan 2002 P4 question paper. The problem is that I need to deduce n(n-1)(2n+5) is divisible by 6 for all n>1. How do I deduce this without induction? This question has only 2 marks.

Thanks in advance.

Alpha:
I wish I could help... :(

Try Yahoo Answers?

elemis:
Let f(n) = n(n-1)(2n+5)

f(1) = 1(1-1)(2+5) = 0  Since zero is divisible by 6 the statement is proven to be divisible for n = 1

Assuming f(k) is divisible by 6 for all k which are positive integers greater than 1.

Therefore, f(k+1) = (k+1)(k)[2(k+1)+5]

f(k+1) - f(k) = (k+1)(k)[2(k+1)+5] - [k(k-1)(2k+5)]

                  = 2k3 +9k2 +7k - 2k3 -3k2 +5k
              
                  =6k2 + 12
 
                  = 6(k2 +2)

Hence,         f(k+1) = 6(k2 +2) + f(k)  is proven to be divisible when n=k+1

If f(n) is divisible when n=k it is shown to be divisible when n=k+1 where k is any positive integer greater than 1.
          

elemis:

--- Quote from: Cleo~patra VII on December 18, 2010, 03:36:35 pm ---I wish I could help... :(

Try Yahoo Answers?

--- End quote ---

No need... Ari is here 8)

Alpha:

--- Quote from: Ari Ben Canaan on December 18, 2010, 03:42:18 pm ---No need... Ari is here 8)

--- End quote ---

Thank you Lion. :)

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