Difference between revisions of "Jadhav Arithmetic Merging Equation"

Latest revision as of 16:45, 14 February 2025

Jadhav Arithmetic Merging Equation

Jadhav Arithmetic Merging Equation or Jadhav's Summation is a formula which can do summation of a finite arithmetic series whose terms are the result of product of respective terms of two arithmetic series with constant common difference (arithmetic progression). Derived by Jyotiraditya Jadhav

Derivation and Statement

Let us consider two different arithmetic progression series.

$a , a+d , a+2d , a+3d ... a+(n-1)d$

$a_1 , a_1+d_1 , a_1+2d_1 , a_1+3d_1 ... a_1+(n-1)d_1$

Now the series formed by product of their respective terms will be

$aa_1 ,(a+d)(a_1+d_1) ,(a+2d)(a_1+2d_1),(a+3d)(a_1+3d_1) ...[a+(n-1)] a_1+(n-1)d_1$

So the summation of all the terms will be

$\Sigma a_1+(a+d)(a_1+d_1) ...+[a+(n-1)d][a_1+(n-1)d_1]$

Giving the Jadhav Arithmetic Merging Equation

$n(a_1a)+d_1d \frac{(n-1)(n)(2n-1)}{6}+[{a_1d+ad_1}]\frac{(n-1)n}2$

Nomenclature

n : Total number of terms in the corresponding series
a : First term of first series
$a_1$ : First term of second series
d : Common difference of first series
$d_1$ : Common difference of second series

History

Been known Jyotiraditya Jadhav was trying to find a generalized formula to easily apply Factorial to any number which was as superior as Binomial_theorem and came across a situation of multivariate binomials and was necessary to find their product to complete Factorial Equation and so started finding direct [1] to complete it but was pretty unsuccessful in it and hence started with algebraic patterns and made Jadhav Arithmetic Merging Equation overnight.

This article has been proposed for deletion. The reason given is: lacks notability. Sysops: Before deleting this article, please check the article discussion pages and history.

@@ Line 1: / Line 1: @@
 [[File:Jadhav Arithmetic Merging Equation.png|thumb|591x591px|Jadhav Arithmetic Merging Equation]]
-'''Jadhav Arithmetic Merging Equation''' or '''Jadhav's Summation''' is a '''Mathematical Formula''' which can do '''summation of a finite arithmetic series''' '''whose terms are the result of product''' of respective terms of two arithmetic series with constant common difference (arithmetic progression). Derived by Indian Mathematician '''[[Jyotiraditya Jadhav|Jyotiraditya Abhay Jadhav.]]'''
+'''Jadhav Arithmetic Merging Equation''' or '''Jadhav's Summation''' is a formula which can do summation of a finite arithmetic series whose terms are the result of product of respective terms of two arithmetic series with constant common difference (arithmetic progression). Derived by  [[Jyotiraditya Jadhav]]
-== Formula ==
+== Derivation and Statement ==
 Let us consider two different arithmetic progression series.
@@ Line 26: / Line 26: @@
 * '''<math>a_1 </math> : First term of second series'''
 * '''d : Common difference of first series'''
 * '''<math>d_1  </math>: Common difference of second series'''
 == History ==
-Been known Jyotiraditya Jadhav was trying to find a '''Generalized formula''' to easily apply '''[[wikipedia:Factorial|factorial function]]''' to any number which was as superior as '''[[wikipedia:Binomial_theorem|Isaac Newton's Binomial Theorem]]''' and came across a situation of multivariate binomials and was necessary to find their product to complete Factorial Equation and so started finding direct [https://www.ck12.org/book/ck-12-middle-school-math-concepts-grade-7/section/1.2/#:~:text=A%20numerical%20pattern%20is%20a,consecutive%20numbers%20in%20the%20sequence. numerical patterns] to complete it but was pretty unsuccessful in it and hence started with '''algebraic patterns''' and made '''Jadhav Arithmetic Merging Equation''' over-night.
+Been known Jyotiraditya Jadhav was trying to find a generalized formula to easily apply [[Factorial]] to any number which was as superior as [[Binomial_theorem]] and came across a situation of multivariate binomials and was necessary to find their product to complete Factorial Equation and so started finding direct [//www.ck12.org/book/ck-12-middle-school-math-concepts-grade-7/section/1.2] to complete it but was pretty unsuccessful in it and hence started with algebraic patterns and made Jadhav Arithmetic Merging Equation overnight.
-Read More about [[Jyotiraditya Jadhav|Jyotiraditya Jadhav here]]
-== '''Practical Observation''' ==
-Let us take series of 1,2,3.....10
-And the second series of 1,3,5.....19
-Now sum of product of respective terms will be using Jadhav Arithmetic Merging Equation gives :
-<math>10(1 \times 1)+1 \times 2 \frac{(10-1)(10)(20-1)}{6}+[{2+1}]\frac{(10-1)10}2 = 430   </math>
-and this works for any arithmetic series in real numbers.
-== Derivation ==
-Let the two arithmetic series be
-<math>a , a+d , a+2d , a+3d ... a+(n-1)d</math>
-<math>a_1 , a_1+d_1 , a_1+2d_1 , a_1+3d_1 ... a_1+(n-1)d_1</math>
-<math>aa_1 ,(a+d)(a_1+d_1) ,(a+2d)(a_1+2d_1),(a+3d)(a_1+3d_1) ...[a+(n-1)] a_1+(n-1)d_1 </math>
-and sum of all the terms of this third series will be
-<math>\Sigma a_1+(a+d)(a_1+d_1) ...+[a+(n-1)d][a_1+(n-1)d_1]   </math>
-Now we can use foil method to open-up brackets and get each term.
-<math>\Sigma a_1a+a_1a+d_1d+a_1d+d_1a ...+a_1a+d_1d+[{a_1d+d_1a}] (n-1)^2   </math>
-Now using laws of algebra and formula of sum of squares of Arithmetic Series we get,
-<math>n(a_1a)+d_1d \frac{(n-1)(n)(2n-1)}{6}+[{a_1d+ad_1}]\frac{(n-1)n}2  </math>
-Hence Deriving and Proving Jadhav Arithmetic Merging Equation.
+{{delete|lacks notability}}

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Difference between revisions of "Jadhav Arithmetic Merging Equation"

Latest revision as of 16:45, 14 February 2025

Derivation and Statement

Nomenclature

History