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My Omnidimensional AP Research: From a Simple Number Pattern to a General Power-Sum Formula

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My Omnidimensional AP Research: From a Simple Number Pattern to a General Power-Sum Formula By Kumaran Kanniappan My OMNIBAL research explores a question: If we know exactly how a sequence is generated, can we calculate its powered total without visiting every term? For arithmetic progressions, we can express that calculation through a midpoint, symmetry, and reusable moment formulas. In this article, I explain the approach from a simple example to multidimensional arrangements and higher powers, then describe how I would evaluate its practical usefulness. 1. BEGIN WITH A FAMILIAR PATTERN Consider: 3, 5, 7, 9, 11, 13, 15, 17 Each number increases by two. This is an arithmetic progression, or AP. Write it as: a_i = F + h*i, for i = 0,...,X-1 The parameters are: - F: first value. - h: common difference. - X: number of values. For this sequence: F = 3, h = 2, X = 8 Its last value is: L = F + h*(X-1) = 17 Its midpoint is: M = (F+L)/2 = 10 2. THE MIDPOINT REVEALS THE STRUCTURE Pair the valu...
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Problem Solving : Cryptarithmatic Problem 1 :Find the value of P from [ A L E * R U M = E R M P N E]

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ALE

 * RUM

 ------------


W I N E

W U W L

E W W E

--------------------
E R M P N E

 ______________
Here is the solution: 0 to 9 , 10 numbers Should be replaced with those following Characters A, L, E, R, U, M, W, I, N, P so 10 characters replaced with 10 digits from 0 to 9 Consider M * ALE = WINE

and U * ALE = WUWL

and R * ALE = EWWE

See in last R *ALE = EWWE

R * E = E will be last digit so it possible only by multiplication of 5 with odd values so 1 * 5 or 

 3* 5 or 7*5 and 9*5 = last unit digit always 5 so E = 5 but 5* 5 not possible because R is not 

 equal to W.....and look at multiplied value N + L = N so any value on addition with 0 will give 

 you result as same hence N + 0 = N then L = 0

so remaining A and R are left R possible values are R = 1, 3, 7, 9 but 1 is not possible because 

 1 * ALE is equal to ALE and R = 3, 7, 9 are possible......

Already we know that E = 5 and L = 0 then

R * A05 = 5WW5 (apply logic here with R = 3, 7 , 5)

for example R =3 means 3 * A05 = (51)15 (EWWE) so 3 not possible because 3 multiple with 

 anyother number ust give result as 51 so it's impossible

R= 7 then 7 *A05 = (53)35 again 7 also not possible because multiplication of 7 never give the 

 value 53

so R= 9 then 9*A05 = (54)45 but look at it 9 is possible because if 6*9 =54 so A= 6 hence then 

 we found all values

9 * 605 = 5445 upto here we know that A= 6 L=0 E=5 R=9 and W = 4 remaining possible digits 

 are (1, 2,3,7,8)

605*9UM
------------
4IN5
4U40
5445
--------------------
5RMPN5

look at U * 605 = 4U40

U*5 =40 so U = 8 hence 8 *605 = 4840 remaining digits are (1,2,3,7)

and look at 605 * M = 4IN5 here M is not 1 because 1 * 605 is 605 but not possible because

 4IN5 as well as 2 is not also not possible because 2 * 5 = (1)0 in unit digit o not possible 

 because 4IN5 hence 3 and 7 are possible so


if M=3


3 * 605 = 1815 = 4IN5 so 3 also not possible because 4IN5 not matched with 1815

hence M=7 then 7 * 605 = 4235 = 4IN5 hence pattern has been matched so M=7 and I = 2 and 

 finally N = 3....

now what is the remaining digit only 1 hence you can quickly answer that (P = 1)

so totally we got A= 6 L=0 E=5 R=9 W = 4 M=7 and I = 2 and finally N = 3

605*987
------------
4235
4840
5445
--------------------
597135

hence here you will get P = 1

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