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, 2 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

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 - 3 years, 2 months ago

Log in to reply

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 - 3 years, 2 months ago

Log in to reply

Haha :), thanks !

Karthik Venkata - 3 years, 2 months ago

Log in to reply

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

Mehul Arora - 3 years, 2 months ago

Log in to reply

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

Karthik Venkata - 3 years, 2 months ago

Log in to reply

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

Mehul Arora - 3 years, 2 months ago

Log in to reply

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

Karthik Venkata - 3 years, 2 months ago

Log in to reply

@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 - 3 years, 2 months ago

Log in to reply

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

Karthik Venkata - 3 years, 2 months ago

Log in to reply

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

Mehul Arora - 3 years, 2 months ago

Log in to reply

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

Karthik Venkata - 3 years, 2 months ago

Log in to reply

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

Mehul Arora - 3 years, 2 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...