Proving a Simple Formula for n = 2,4,6,8 Using Mathematical Induction

In summary, mathematical induction is a proof technique used to show that a statement is true for all natural numbers. It differs from other proof techniques in that it is specific to natural numbers and relies on the principle of strong induction. The principle of strong induction allows for more assumptions in the proof, making it more powerful than weak induction. Mathematical induction can be used to prove statements about natural numbers, sequences, and series, but not for statements involving real numbers or infinite sets.
  • #1
duki
264
0

Homework Statement



Conjecture a simple formula for n = 2,4,6,8

Homework Equations



Kind of...
[tex]Tn = (1-\frac{1}{n^2})(1-\frac{1}{(n-1)^2})... to n=1[/tex]

The Attempt at a Solution



The pattern is 3/4, 5/8, 7/12, 9/16
I know that the top is increasing by 2 and the bottom is increasing by 4... but I don't know how to put that into a formula to which I can prove.

Thanks for any help!
 
Last edited:
Physics news on Phys.org
  • #2
oooooh snap, how about this:

[tex]Tn = \frac{n+1}{n*2}[/tex] for all even n?
 
  • #3
Why not?
 

FAQ: Proving a Simple Formula for n = 2,4,6,8 Using Mathematical Induction

What is mathematical induction?

Mathematical induction is a method of proving that a statement is true for all natural numbers by first proving it is true for the first natural number, then showing that if it is true for any natural number, it must also be true for the next natural number.

How is mathematical induction different from other mathematical proof techniques?

Mathematical induction is used specifically for proving statements about natural numbers, while other proof techniques may be used for more general types of statements. Induction also relies on the principle of strong induction, which allows for more assumptions in the proof.

What is the principle of strong induction?

The principle of strong induction states that if a statement is true for all natural numbers up to and including some number n, and if the statement holds for n+1 whenever it holds for all natural numbers up to and including n, then the statement is true for all natural numbers.

What is the difference between strong induction and weak induction?

Weak induction only requires the base case (proving the statement is true for the first natural number) and the inductive step (proving that if the statement is true for n, it is also true for n+1). Strong induction, on the other hand, allows for additional assumptions in the inductive step, making it a more powerful proof technique.

What types of statements can be proven using mathematical induction?

Mathematical induction is typically used to prove statements about natural numbers, such as number patterns, divisibility, and inequalities. It can also be used to prove statements about sequences and series. However, it cannot be used for statements involving real numbers or infinite sets.

Similar threads

[L'Hospital's Rule] Can I Use Mathematical Induction to Prove This?
Replies
6
Views
1K
Proof by mathematical induction
Replies
4
Views
1K
Proofs By Induction: Proving P(n) for All n
Replies
3
Views
2K
Strong Induction: Proving Every Int n>1 is a Product of Primes
Replies
5
Views
2K
How to Use Mathematical Induction to Prove Equations?
Replies
23
Views
2K
Proof by Induction of String exponentiation? (Algorithms)
Replies
5
Views
1K
Compute sum (possibly using Parseval's formula)
Replies
1
Views
954
Mathematical induction inequality
Replies
4
Views
2K
Proof by Induction of shortest suffix of concatenated string
Replies
1
Views
1K
How Does Sum of Consecutive Numbers Relate to Cubes of Integers?
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top