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Old 9th July 2012, 01:42 AM   #1 (permalink)
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question 1

show that [ n(n+3) ] / [ 4(n+1)(n+2) ] < 1/4 for all positive integers of n
whats the proper step of doing this?

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Old 9th July 2012, 11:29 AM   #2 (permalink)
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Re: question 1

We could use mathematical induction, but i prefer long division.
[ n(n+3) ] / [ 4(n+1)(n+2) ] = 1/4 - 2/[(n+1)(n+2)] ultimately,
So as 2/[(n+1)(n+2)] >0 as n is a positive integer;

[ n(n+3) ] / [ 4(n+1)(n+2) ] = 1/4 - 2/[(n+1)(n+2)]< 1/4

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Old 9th July 2012, 08:01 PM   #3 (permalink)
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Re: question 1

You can also multiply both sides by 4(n+1)(n+2).

LHS*4(n+1)(n+2)=n˛+3n
RHS*4(n+1)(n+2)=n˛+3n+2

4(n+1)(n+2)>0 since n is positive.
Hence from the result above, LHS*4(n+1)(n+2)<RHS*4(n+1)(n+2) → LHS<RHS

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