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Induction practice for beginners

Using induction, prove that

\[\left(1 - \dfrac {2}{4}\right)\left(1 - \dfrac {2}{5}\right)\ldots\left(1 - \dfrac{2}{n}\right) = \dfrac{6}{n(n-1)}\]

for \(n \geq 4\).

Note by Sharky Kesa
2 years, 2 months ago

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PROOF BY INDUCTION

Let \( T(n) \) be the proposition that \( \forall n \geq 4 \), we have \[\left(1 - \dfrac {2}{4}\right)\left(1 - \dfrac {2}{5}\right)\ldots\left(1 - \dfrac{2}{n}\right) = \dfrac{6}{n(n-1)}\]

Base Case :- \( T(4) \) is true because \( (1 - \dfrac{2}{4}) = \dfrac{6}{4(4-1)} \)

Inductive Step :- Let \( T(k) \) be true for some \( k \geq 4 \), that is -

\[\left(1 - \dfrac {2}{4}\right)\left(1 - \dfrac {2}{5}\right)\ldots\left(1 - \dfrac{2}{k}\right) = \dfrac{6}{k(k-1)}\]

Multiplying both sides of the above equation by \( (1 - \dfrac{2}{k+1}) \), we get that

LHS of \( T(k+1) = ( \dfrac{6}{k(k-1)}) \times (1 - \dfrac{2}{k+1}) = \dfrac{6}{k(k+1)} = \) RHS of \( T(k+1) \).

Hence, as \( T(k) \) true \( \Rightarrow T(k+1) \) true, our induction step is now complete.

Therefore, by First Principle of Mathematical Induction, we now conclude that \( T(n) \) is true \( \forall n \geq 4 \). Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata Yeah. I learnt Induction today! What a coincidence @Sharky Kesa. And, Flawless proof Karthik Venkata . I solved it using the same way. And also Got it! Happy Dance Mehul Arora · 2 years, 2 months ago

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@Mehul Arora Haha :), thanks ! Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata No need to thank me, Genius :) Mehul Arora · 2 years, 2 months ago

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@Mehul Arora Lol good joke ! By the way, you are of Class 10 too ? Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata No brother, I am class 9 :) Btw, That was not a joke. :P Mehul Arora · 2 years, 2 months ago

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@Mehul Arora So just entered class 9 right ? Nice, you are really talented.. You plan to give RMO ? Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata Yeah, I just entered. And No, I am not That Talented. There are People Much Smarter and intelligent Than me. Some of them Would include @Archit Boobna , @Rajdeep Dhingra and Many more.

I do plan to Give the RMO. Any Tips or Tricks? They Would be of great help! Mehul Arora · 2 years, 2 months ago

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@Mehul Arora No idea, I too am gonna write the RMO for the first time... Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata Okay! Let's compete xD xD ALthough I am sure you will win :P xD Mehul Arora · 2 years, 2 months ago

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@Mehul Arora Haha, hope we meet in person at the INMO Camp next year :P ! Karthik Venkata · 2 years, 2 months ago

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@Karthik Venkata Yeah, Sure! I sure want to Meet a genius in person xD Mehul Arora · 2 years, 2 months ago

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