User:RakeshPrem001/Jadhav Theorem

Jadhav Theorem or Jadhav Arithmetic theorem is a equation which is applicable for any 3 terms of an Arithmetic Progression with a constant common difference between them. This theorem is derived by Jyotiraditya Jadhav. (made by)

Statement
If any three consecutive numbers are taken say a,b and c with a constant common difference, then the difference between the square of the 2nd term (b) and the product of the first and the third term (ac) will always be the square of the common difference (d).

Representation of statement in variable :

B2 - ac = d2

ɞ2 - ɑɣ = $$\delta$$2

Nomenclature

 * [[File:Поиск одноцветной прогрессии в доказательстве канонической теоремы ван дер Вардена.gif|thumb|Arithmetic Progressions]]a /$$\alpha$$ : First term of Arithmetic Series
 * B/$$\beta$$: second / Middle term of series
 * c/$$\gamma$$: third/ last term of series
 * d/$$\delta$$: Common difference of the arithmetic progression
 * c/$$\gamma$$: third/ last term of series
 * d/$$\delta$$: Common difference of the arithmetic progression
 * d/$$\delta$$: Common difference of the arithmetic progression

Practical Observation
• Let, a be 1, b be 2 and c be 3 (Arithmetic progression with common difference of 1)

By equating in the formula

(2)2 – (1 x 3) = 1 (Square of 1/-1)

• Let, a be 10, b be 20 and c be 30 (Arithmetic progression with common difference of 10)

By equating in the formula

(20)2 – (10 X 30) = 100 (square of 10/-10)

And this is true till endless number series.

Proof
(The following proof is derived by Jyotiraditya Jadhav)

B2 -ac

= (a)2 – (a-d)(a+d)                                      (as three terms are in Arithmetic progression)

= A2 – a2 + d2

= (d)2

= d2

Other forms of Theorem
Finding square of Middle number :

ɞ2 = $$\delta$$2 +ɑɣ ---1

Common difference can also be written as :

$$\surd$$ɞ2 - ɑɣ = $$\delta$$  ---2

The product of first term and third term as the negative of difference of square of common difference and the square of second (middle) term:

ɑɣ = -($$\delta$$2 -ɞ2) 3

Uses
• This can be used in daily life to find square of any number (mentally) as we can better explain

with a example :

Lets find square of 102, so now we can assume this number a part of a arithmetic series

Let the series be 100, 102 and 104 where common difference is 2

Now we can derive the following with the given formula

B2 = d2 + AC (from 1)

So now the square of common difference is 4 and the product of A(100) and C (104) can be

written as 104 X 100 and now the product of 104 and 100 can be found easily mentally as

10400 and later adding square of common difference (4) into it will make it 10404 and that

is square of 102.

This will be easy to understand :

1. Lets find square of 406

2. So it can be term of arithmetic progression 400,406,412 (common difference = 6)

3. Now 400 X 412 can be easily found mentally as 164800 and later adding square of common

difference (36) to it makes it 164836 which is square of 406

• This pattern can be used to make equations for unknown quantities of the arithmetic series

as this is in a form of 4 variables then it can be used to make a equation of 4 unknown

quantities with other three equations (quadratic equation).