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)
Jadhav Theorem | |
---|---|
Arithmetic Progressions Theorem | |
Definition | |
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. | |
Details | |
Applications | Arithmetic Progressions, Quadratic Equations, Gamma Function of Hoyrop J. etc. |
Indian Invention |
Statement[edit]
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 = 2
Nomenclature[edit]
- a / : First term of Arithmetic Series
- B/: second / Middle term of series
- c/: third/ last term of series
- d/: Common difference of the arithmetic progression
Practical Observation[edit]
• 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[edit]
(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[edit]
Finding square of Middle number :
ɞ2 = 2 +ɑɣ -------------------1
Common difference can also be written as :
ɞ2 - ɑɣ = -------------------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 = -(2 -ɞ2) ----------------3
Uses[edit]
• 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).[1]
See also[edit]
References[edit]
This article "Jadhav theorem" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Jadhav theorem. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.