IGCSE/GCSE/O & A Level/IB/University Student Forum

Qualification => Subject Doubts => GCE AS & A2 Level => Math => Topic started by: tantrik on December 18, 2010, 03:06:38 pm

Title: Help! cannot solve P4 problem
Post by: tantrik on December 18, 2010, 03:06:38 pm: 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.
Title: Re: Help! cannot solve P4 problem
Post by: Alpha on December 18, 2010, 03:36:35 pm: I wish I could help... :(

Try Yahoo Answers?
Title: Re: Help! cannot solve P4 problem
Post by: elemis on December 18, 2010, 03:41:57 pm: 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.
Title: Re: Help! cannot solve P4 problem
Post by: elemis on December 18, 2010, 03:42:18 pm: Quote from: Cleo~patra VII on December 18, 2010, 03:36:35 pm
I wish I could help... :(

Try Yahoo Answers?

No need... Ari is here 8)
Title: Re: Help! cannot solve P4 problem
Post by: Alpha on December 19, 2010, 02:26:20 am: Quote from: Ari Ben Canaan on December 18, 2010, 03:42:18 pm
No need... Ari is here 8)

Thank you Lion. :)
Title: Re: Help! cannot solve P4 problem
Post by: tantrik on December 19, 2010, 08:47:32 am: Thanks Ari. Highly appreciated. :)
Title: Re: Help! cannot solve P4 problem
Post by: elemis on December 19, 2010, 09:07:34 am: Quote from: tantrik on December 19, 2010, 08:47:32 am
Thanks Ari. Highly appreciated. :)

No worries.