Proof Question: Using Mathematical Induction

Click For Summary

Discussion Overview

The discussion revolves around a homework problem requiring the proof of a mathematical statement using induction. The specific statement involves a summation of fractions and aims to establish an equality for all integers n greater than or equal to 1.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant presents the statement to be proven, which involves a summation of fractions equating to a specific expression.
  • Another suggests using partial fractions to express the left-hand side as a telescoping series.
  • Several participants express confusion about the use of partial fractions and seek clarification on the steps involved.
  • A participant proposes naming the sum to simplify the induction proof process.
  • There are multiple requests for help and clarification on how to manipulate the expressions and perform algebraic simplifications.
  • Participants discuss the steps of the proof, including how to handle polynomial fractions and the implications of certain algebraic manipulations.
  • One participant points out a potential misprint in the original statement, prompting further clarification.
  • There is ongoing uncertainty about specific algebraic steps and simplifications, with participants asking for guidance on how to proceed.

Areas of Agreement / Disagreement

Participants generally agree on the need to prove the statement using induction, but there is significant uncertainty and confusion regarding the specific algebraic manipulations and the use of partial fractions. No consensus is reached on the best approach to simplify the expressions involved.

Contextual Notes

Participants express varying levels of familiarity with the concepts of summation, partial fractions, and polynomial fractions, indicating a range of understanding that affects the discussion.

  • #31
So, what i need to do is:

\frac{n(n+2)}{(n+1)(n+2)} + \frac{1}{(n+1(n+2)}

Doesnt that just equal:

\frac{n(n+2)}{(n+1)(n+2)}?
 
Physics news on Phys.org
  • #32
… attack of the clones …

Lococard2 said:
So, what i need to do is:

\frac{n(n+2)}{(n+1)(n+2)} + \frac{1}{(n+1(n+2)}

Doesnt that just equal:

\frac{n(n+2)}{(n+1)(n+2)}?

Hey … what happened to the original Lococard? :confused:

I liked him!

:redface: erm … just read what you've written … :redface:​
 
  • #33
haha, Sorry i lost my password and this connection doesn't allow gmail.com to load.

But I am back now.

\frac{n(n+2)+1}{(n+1)(n+2)}

And from then i simplfy it?
 
  • #34
… long live lococard … !

Lococard said:
haha, Sorry i lost my password and this connection doesn't allow gmail.com to load.

But I am back now.

I can tell the difference! :biggrin:
\frac{n(n+2)+1}{(n+1)(n+2)}

And from then i simplfy it?

Yes! :smile:
 
  • #35
So that is the same as:

\frac{(n+1)(n+1)}{(n+1)(n+2)}?

yes? no?

Now I am not sure what to do :S
 
  • #36
Lococard said:
So that is the same as:

\frac{(n+1)(n+1)}{(n+1)(n+2)}?

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 \frac{(n+1)}{(n+1)}\,\frac{(n+1)}{(n+2)} , which is … ? :smile:
 
  • #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 … !

Schrödinger's Dog said:
= \frac{(n+2) - 1} {(n+2)}? = 1-\frac{1}{n+1} \equiv \sum_{k=1}^n \frac{1}{k(k+1)}​

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 \neq 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. :-pEDIT: 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:

Similar threads

Mathematica Solve equation perturbatively
  • · Replies 2 ·
Replies
2
Views
2K
Mathematica Solving equation and plotting issue with Mathematica
  • · Replies 2 ·
Replies
2
Views
2K
Mathematica How can I rederive this Mathematica result for a complex equation?
  • · Replies 2 ·
Replies
2
Views
2K
[L'Hospital's Rule] Can I Use Mathematical Induction to Prove This?
  • · Replies 6 ·
Replies
6
Views
2K
Mathematica Series expansion from the red book on special functions by Richard Ask
  • · Replies 4 ·
Replies
4
Views
1K
Mathematica Plotting a parameter Plane of a spring pendulum in Mathematica
  • · Replies 1 ·
Replies
1
Views
2K
Mathematica Solve complex equation analytically
  • · Replies 19 ·
Replies
19
Views
3K
Mathematica Solving a complicated equation for approximate analytical Solution using Mathematica
  • · Replies 10 ·
Replies
10
Views
3K
Proof by induction for rational function
  • · Replies 7 ·
Replies
7
Views
2K
Induction with binomial coefficient
  • · Replies 15 ·
Replies
15
Views
3K