Induction practice for beginners

Using induction, prove that

(124)(125)(12n)=6n(n1)\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 n4n \geq 4.

Note by Sharky Kesa
4 years, 3 months ago

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PROOF BY INDUCTION

Let T(n) T(n) be the proposition that n4 \forall n \geq 4 , we have (124)(125)(12n)=6n(n1)\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) T(4) is true because (124)=64(41) (1 - \dfrac{2}{4}) = \dfrac{6}{4(4-1)}

Inductive Step :- Let T(k) T(k) be true for some k4 k \geq 4 , that is -

(124)(125)(12k)=6k(k1)\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 (12k+1) (1 - \dfrac{2}{k+1}) , we get that

LHS of T(k+1)=(6k(k1))×(12k+1)=6k(k+1)= T(k+1) = ( \dfrac{6}{k(k-1)}) \times (1 - \dfrac{2}{k+1}) = \dfrac{6}{k(k+1)} = RHS of T(k+1) T(k+1) .

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

Therefore, by First Principle of Mathematical Induction, we now conclude that T(n) T(n) is true n4 \forall n \geq 4 .

Karthik Venkata - 4 years, 3 months ago

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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 - 4 years, 3 months ago

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Haha :), thanks !

Karthik Venkata - 4 years, 3 months ago

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@Karthik Venkata No need to thank me, Genius :)

Mehul Arora - 4 years, 3 months ago

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@Mehul Arora Lol good joke ! By the way, you are of Class 10 too ?

Karthik Venkata - 4 years, 3 months ago

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@Karthik Venkata No brother, I am class 9 :) Btw, That was not a joke. :P

Mehul Arora - 4 years, 3 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 - 4 years, 3 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 - 4 years, 3 months ago

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@Mehul Arora No idea, I too am gonna write the RMO for the first time...

Karthik Venkata - 4 years, 3 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 - 4 years, 3 months ago

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@Mehul Arora Haha, hope we meet in person at the INMO Camp next year :P !

Karthik Venkata - 4 years, 3 months ago

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@Karthik Venkata Yeah, Sure! I sure want to Meet a genius in person xD

Mehul Arora - 4 years, 3 months ago

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