Proof Question: Using Mathematical Induction

In summary, for all integers n =>1, the sum of the series from 1 to n+1 of 1/(k(k+1)) is equal to 1-1/(n+1). This can be proven using partial fractions and the induction method
  • #36
Lococard said:
So that is the same as:

[itex]\frac{(n+1)(n+1)}{(n+1)(n+2)}?[/itex]

yes? no?

Yes! :smile:

(You knew that, didn't you?)
Now I am not sure what to do :S

You really don't dig polynomial fractions, do you? :frown:

It's the same as [itex]\frac{(n+1)}{(n+1)}\,\frac{(n+1)}{(n+2)}[/itex] , which is … ? :smile:
 
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  • #37
Im really not sure.

You can cancel the (n+1) / (n+1)


Leaving (n+1) / (n+2)?



Im a little confused as the teacher suggested that i work my way to get 1 - [1 / (n+2)]
 
  • #38
Lococard said:
You can cancel the (n+1) / (n+1)

Leaving (n+1) / (n+2)?

Yes! :smile:
Im a little confused as the teacher suggested that i work my way to get 1 - [1 / (n+2)]

Well, maybe teacher is right …

What is 1 - [1 / (n+2)] ?

Hint: 1 = … ? :smile:
 
  • #39
(n+2) / (n+2) - [1/(n+2)]?

= [(n+2) - 1] / (n+2)?
 
  • #40
Yes!

= … ? :smile:
 
  • #41
… merveilleux … !

Schrodinger's Dog said:
[tex]= \frac{(n+2) - 1} {(n+2)}? = 1-\frac{1}{n+1} \equiv \sum_{k=1}^n \frac{1}{k(k+1)}[/tex]​

erm …
:redface: … êtes-vous sûr …? :redface:
 
  • #42
tiny-tim said:
erm …
:redface: … êtes-vous sûr …? :redface:

Which of those equals the last one, I left the question marks in there for a good reason, writing which of these is correct in latex gets messy, Ie ?=. The point is that one of those = a third, or the summation given at the beginning of the thread with a telescoping series.

Ie you have your answer. Et voila...

Since obviously some people found this confusing I've edited it to clear that up.

I kept putting [itex]\neq[/itex] in then removing it then putting it in again before I decided it looks better with the question marks. :smile:

If you don't believe me ask a mentor if I edited it about 219283927^34839 times. :tongue:EDIT: Actually sod it I might as well delete it, as it only makes sense, if you take on board the post about 1/k(k+1) being equivalent to the LHS. So nm.

I was trying to helpfully point out that what gib7 said was the same as what you eventually end up with, but it came out completely wrong.
 
Last edited:
<h2>1. What is mathematical induction?</h2><p>Mathematical induction is a proof technique used to prove statements that involve natural numbers or integers. It involves proving that a statement is true for a base case, and then showing that if the statement is true for a particular number, it is also true for the next number.</p><h2>2. How do you use mathematical induction to prove a statement?</h2><p>To use mathematical induction, you start by proving the statement is true for the base case, typically n = 1. Then, you assume that the statement is true for some arbitrary number n, and use this assumption to prove that the statement is also true for n+1. This shows that if the statement is true for n, it is also true for n+1. Finally, you use the principle of mathematical induction to conclude that the statement is true for all natural numbers.</p><h2>3. What types of statements can be proved using mathematical induction?</h2><p>Mathematical induction is typically used to prove statements about natural numbers or integers. This can include statements about sequences, series, and divisibility, as well as more complex statements involving algebraic expressions or inequalities.</p><h2>4. What is the difference between weak and strong induction?</h2><p>Weak induction, also known as the principle of mathematical induction, involves proving that a statement is true for a base case and then using this to show that it is true for the next number. Strong induction, on the other hand, involves proving that a statement is true for all numbers up to a given number, and then using this to show that it is also true for the next number. Strong induction is a more powerful form of induction, but both methods are valid and can be used to prove statements.</p><h2>5. Are there any common mistakes when using mathematical induction?</h2><p>One common mistake when using mathematical induction is assuming that a statement is true for a particular number without actually proving it. Another mistake is using the wrong base case or incorrectly applying the induction hypothesis. It is important to carefully follow the steps of mathematical induction and make sure that each part of the proof is valid.</p>

1. What is mathematical induction?

Mathematical induction is a proof technique used to prove statements that involve natural numbers or integers. It involves proving that a statement is true for a base case, and then showing that if the statement is true for a particular number, it is also true for the next number.

2. How do you use mathematical induction to prove a statement?

To use mathematical induction, you start by proving the statement is true for the base case, typically n = 1. Then, you assume that the statement is true for some arbitrary number n, and use this assumption to prove that the statement is also true for n+1. This shows that if the statement is true for n, it is also true for n+1. Finally, you use the principle of mathematical induction to conclude that the statement is true for all natural numbers.

3. What types of statements can be proved using mathematical induction?

Mathematical induction is typically used to prove statements about natural numbers or integers. This can include statements about sequences, series, and divisibility, as well as more complex statements involving algebraic expressions or inequalities.

4. What is the difference between weak and strong induction?

Weak induction, also known as the principle of mathematical induction, involves proving that a statement is true for a base case and then using this to show that it is true for the next number. Strong induction, on the other hand, involves proving that a statement is true for all numbers up to a given number, and then using this to show that it is also true for the next number. Strong induction is a more powerful form of induction, but both methods are valid and can be used to prove statements.

5. Are there any common mistakes when using mathematical induction?

One common mistake when using mathematical induction is assuming that a statement is true for a particular number without actually proving it. Another mistake is using the wrong base case or incorrectly applying the induction hypothesis. It is important to carefully follow the steps of mathematical induction and make sure that each part of the proof is valid.

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