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Difference between revisions of "Jadhav Theorem"

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'''Jadhav Theorem''', or '''Jadhav Arithmetic Theorem''' is an equation which is applicable for any three consecutive terms of an [[arithmetic sequence]]. This theorem is derived by [[Jyotiraditya Jadhav]].
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'''Jadhav Theorem''', or '''Jadhav Arithmetic Theorem''', is a theorem derived by [[Jyotiraditya Jadhav]] which is applicable for any three consecutive terms of an [[arithmetic sequence]].
  
 
== Statement ==
 
== Statement ==
  
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If any three numbers <math>a</math>, <math>b</math> and <math>c</math> are taken 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). In other words, <math>b^2-ac = d^2</math>
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Let <math>a</math>, <math>b</math> and <math>c</math> be three consecutive terms in an arithmetic sequence with common difference <math>d</math>. Then <math>b^2-ac = d^2</math>
  
 
== Proof ==
 
== Proof ==
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* This can be used to find the square of any number without a calculator.
 
* This can be used to find the square of any number without a calculator.
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** For example, let's find square of <math>102</math>. This number is part of the arithmetic series <math>100,102,104</math> with common difference <math>2</math>. Wwe can derive <math>b^2 = d^2+ac</math> from Jadhav Theorem. The square of common difference is 4 and the product of <math>a = 100</math> and <math>c = 104</math> is 10400 and later adding square of common difference (4) into it will make it 10404 and that is square of 102.
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** For example, let's find square of <math>102</math>. This number is part of the arithmetic series <math>100,102,104</math> with common difference <math>2</math>. We can derive <math>b^2 = d^2+ac</math> from Jadhav Theorem. The square of common difference is 4 and the product of <math>a = 100</math> and <math>c = 104</math> is 10400 and later adding square of common difference (4) into it will make it <math>10404</math>, so <math>102^2=10404.</math>
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** Let's find the square of 406. It is a term of arithmetic progression <math>400,406,412</math> with common difference 6. <math>400 \cdot 412</math> can be easily found as <math>164800</math>, and adding square of common difference (36) to this makes it 164836 which is square of 406.
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** Let's find the square of 406. It is a term of arithmetic progression <math>400,406,412</math> with common difference 6. <math>400 \cdot 412</math> can be easily found as <math>164800</math>, and adding square of common difference (36) to this 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).
 
* 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).
  
 
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Revision as of 16:15, 14 February 2025

Jadhav Theorem, or Jadhav Arithmetic Theorem, is a theorem derived by Jyotiraditya Jadhav which is applicable for any three consecutive terms of an arithmetic sequence.

Statement

Let $a$, $b$ and $c$ be three consecutive terms in an arithmetic sequence with common difference $d$. Then $b^2-ac = d^2$

Proof

From difference of squares, the equation $b^2-d^2 = (b-d)(b+d)$ holds. We can rewrite $b-d$ and $b+d$ as $a$ and $c$, respectively. Now our equation is $b^2-d^2 = ac$, and rearranging gives us $b^2-ac = d^2$, as desired.

Uses

  • This can be used to find the square of any number without a calculator.
    • For example, let's find square of $102$. This number is part of the arithmetic series $100,102,104$ with common difference $2$. We can derive $b^2 = d^2+ac$ from Jadhav Theorem. The square of common difference is 4 and the product of $a = 100$ and $c = 104$ is 10400 and later adding square of common difference (4) into it will make it $10404$, so $102^2=10404.$
    • Let's find the square of 406. It is a term of arithmetic progression $400,406,412$ with common difference 6. $400 \cdot 412$ can be easily found as $164800$, and adding square of common difference (36) to this 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).

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