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Topic: D.A.P series - a new sequence of numbers
Replies: 4   Last Post: Apr 27, 2013 9:58 AM

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Doctor Nisith Bairagi

Posts: 23
From: Uttarpara, West Bengal, India
Registered: 3/2/13
Re: D.A.P series - a new sequence of numbers
Posted: Apr 24, 2013 10:23 AM
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From: Doctor Nisith Bairagi
Subject: DAP Series ? a new sequence of numbers
(posted on April 20,2013 by Shyamal Kumar Das, West Bengal, India)
Date: April 24, 2013
(alt.math.recreational.independent)
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Dear Shyamal Babu,
I have gone through your Topic on DAP keenly, and verified the arithmetic of the Problems provided.

(1) You have identified a class of AP sequence, the difference between the consecutive numbers of which is in AP. You want to call this class as: Delayed Arithmetic Progression (DAP) Series.
Why <delayed> is not so clear.

(2) You have overlooked that: (1, 3, 5, 7, ), is a Sequence, and: (1 + 3 + 5 + 7 + ), is a Series. Flatly you have used the term Series, irrespective of Sequence or Series.

(3) The rule that: (D-A) = 3(C-B), though works well, but is obvious, and does not deserve to get special mention.

(4) You have used directly the standard solution of the sum:
S<n> = 1^2 + 2^2 + 3^2 + etc = n(n + 1)(2n + 1)/6.
The formula for this derivation is a standard text book one, and utilized from:
n^3-(n-1)^3 = 3(n-1)^2 + 3(n-1) + 1.
If you could formulate an Alternative Method of finding the sum:
S<n> = 1^2 + 2^2 + 3^2 + etc, on the basis of your (proposed) Salient Features of DAP Series, it would then be called, a CONTRIBUTON.
At present, the Topic appears to be a stereotype school level math exercise problem.

(5) Perhaps this class of DAP can be treated as Recurring sequence /series. A variety of such similar typical special problems for t<n> and S<n>, have already been discussed with the help of <scale of relation> concept. [Reference: Higher Algebra by Hall & Knight (1891), Chap XXIV, McMillan Co].
As I understand that, College Algebra, and Higher Algebra Books deal with similar typical complicated and harder problems. They might reveal more information in this regard.

(6) I appreciate your good amount of intelligent effort in isolating and identifying this class of Sequence / Series, as DAP. For this, credit is due to your effort.

You have energized me to investigate further in this direction for opening a different gate, in generalised way.

Accept my thanks.

Doctor Nisith Bairagi
Uttarpara, West Bengal, India.
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