# 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
3 years, 8 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 $$\forall n \geq 4$$.

- 3 years, 8 months ago

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

- 3 years, 8 months ago

Haha :), thanks !

- 3 years, 8 months ago

No need to thank me, Genius :)

- 3 years, 8 months ago

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

- 3 years, 8 months ago

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

- 3 years, 8 months ago

So just entered class 9 right ? Nice, you are really talented.. You plan to give RMO ?

- 3 years, 8 months ago

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!

- 3 years, 8 months ago

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

- 3 years, 8 months ago

Okay! Let's compete xD xD ALthough I am sure you will win :P xD

- 3 years, 8 months ago

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

- 3 years, 8 months ago

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

- 3 years, 8 months ago