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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]].
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| − | == Statement ==
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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>
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| − | == Proof ==
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| − | From [[difference of squares]], the equation <math>b^2-d^2 = (b-d)(b+d)</math> holds. We can rewrite <math>b-d</math> and <math>b+d</math> as <math>a</math> and <math>c</math>, respectively. Now our equation is <math>b^2-d^2 = ac</math>, and rearranging gives us <math>b^2-ac = d^2</math>, as desired.
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| − | == Uses ==
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| − | * 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>. 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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| − | * 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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