My Math Forum Sequence problems

 Algebra Pre-Algebra and Basic Algebra Math Forum

May 15th, 2015, 03:56 AM   #1
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Sequence problems

Hi, I am having difficulty with these 3 problems. Could anyone give me a hand? Thanx for your help
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 sequence 1.jpg (34.2 KB, 9 views) sequence 2.jpg (25.7 KB, 7 views) sequence 3.jpg (25.4 KB, 8 views)

 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 - (2k-1)^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. Thanks from matisolla
 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/((k-1)(k)(k+1)) = 1/(2(k-1)) - 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. Thanks from matisolla
May 29th, 2015, 03:07 AM   #4
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Quote:
 Originally Posted by skipjack 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/((k-1)(k)(k+1)) = 1/(2(k-1)) - 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.
Thankx skipjack! I could solve the first two. However I dont understand what you explained for the third one. Can you show me how you would solve it? Thanks for helping me so often.

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