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May 15th, 2015, 03:56 AM  #1 
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Sequence problems
Hi, I am having difficulty with these 3 problems. Could anyone give me a hand? Thanx for your help

May 15th, 2015, 05:40 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
(2k)^3  (2k1)^3 = 12*k^2  6*k + 1, so you can rewrite the first as $$ \sum_{k=1}^m 12k^2  6k + 1. $$ This is equal to a cubic polynomial, and you can find it pretty easily if you know the sum formulas. 
May 15th, 2015, 06:18 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,976 Thanks: 2228 
For the first, the sum is 4m³ + 3m², which can be obtained by the method already suggested or proved from scratch by mathematical induction. For the second, use 1/((k1)(k)(k+1)) = 1/(2(k1))  1/k + 1/(2(k+1)) to get a telescoping sum. For the third, if S denotes the required sum, relate 3S  S to a geometric series. 
May 29th, 2015, 03:07 AM  #4  
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Quote:
 

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