You are Here: Home >< Maths

# Proof of something concerning sqrt(2), sqrt(3) and sqrt(5) watch

1. proof that

sqrt(2), sqrt(3) and sqrt(5) cannot all be part of the same geometric progression

i will define sq(2) to be the first term

then let the common difference of the arithmetic progression be d

then

sq(3)=sq(2)+md and sq(5)=sq(2)+nd where (m,n) are posotive integers

so then

(sq(3)-sq(2))/m=(sq(5)-sq(3))/n

and so

sq(3)(m+n)=msq(5)+nsq(2) now i square this then the LHS is an integer where as the RHS is irrational hence we arrive at a contradiction, so all three terms cannot be part of the same Arithmetic progression.

Is this correct?
2. Doesn't the question ask to show they cannot be part of the same geometric progression?
3. I know this isn't particularly helpful, but could you please use LaTeX for this sort of stuff?
4. geometric progression therefore
root(2)=a
root(3)=ar
root(5)=ar^2

therefore since this isn't true (shown by multiplying both sides by root 2root3) there exists no r satisfying the progression.
5. (Original post by Totally Tom)
geometric progression therefore
root(2)=a
root(3)=ar
root(5)=ar^2

therefore since this isn't true (shown by multiplying both sides by root 2root3) there exists no r satisfying the progression.
It doesn't need to be consecutive does it? Then:

where n and m are integers from 0-inf.

That vaguely right?
6. (Original post by Totally Tom)
geometric progression therefore
root(2)=a
root(3)=ar
root(5)=ar^2

therefore since this isn't true (shown by multiplying both sides by root 2root3) there exists no r satisfying the progression.
What if is the 50th or 75th or 3000th term instead of the 2nd term, as you assume here?
7. (Original post by The Bachelor)
What if is the 50th or 75th or 3000th term instead of the 2nd term, as you assume here?
jup, just realised.
8. no the question says arithmetic
9. (Original post by psanghaLFC)
proof that

sqrt(2), sqrt(3) and sqrt(5) cannot all be part of the same geometric progression
wat.
10. whoops, it should be arithmetic progression, but i suppose i could just try and prove both
11. Let sq(3)=r^nsq(2) and sq(5)=r^msq(3) then

(sq(3)/sq(2))^(1/n)=(sq(5)/sq(3))^1/m

so

sq(3)^m/sq(2)^m=sq(5)^n/sq(3)^n

so that would mean

sq(3)^(m+n)=sq(2)^n*sq(5)^m

so after squaring both sides we would see two cases

LHS rational and RHS rational
LHS rational and RHS irrational

in case two it leads to a contradiction, and in case 1 the only prime factors of the RHS are 2 and 5 and of the LHS they are 3 which cannot be the case.
12. (Original post by psanghaLFC)
Let sq(3)=r^nsq(2) and sq(5)=r^msq(3) then

(sq(3)/sq(2))^(1/n)=(sq(5)/sq(3))^1/m

so

sq(3)^m/sq(2)^m=sq(5)^n/sq(3)^n

so that would mean

sq(3)^(m+n)=sq(2)^n*sq(5)^m

so after squaring both sides we would see two cases

LHS rational and RHS rational
LHS rational and RHS irrational

in case two it leads to a contradiction, and in case 1 the only prime factors of the RHS are 2 and 5 and of the LHS they are 3 which cannot be the case.
Seeing as no 'expert' has replied, I may as well have a pop at this proof...

x and y are integer values.

Square both sides

Since and must be integers, are integers. Thus is irrational. Hence the left hand side is always irrational while the right hand side is always rational, hence the equality cannot hold. Proof by contradiction?

Edit: sorry about that, I admit I was stupid enough to overlook his first solution. Ignore, or if anything, it makes it easier to read.
13. yes, thats what he did in his first post

it's just he doesn't latex anything so its horrible to read through-most people don't bother.
14. (Original post by Totally Tom)
yes, thats what he did in his first post

it's just he doesn't latex anything so its horrible to read through-most people don't bother.
lol

wasted 10 minutes of my life!
15. (Original post by Totally Tom)
yes, thats what he did in his first post

it's just he doesn't latex anything so its horrible to read through-most people don't bother.
Haha, my point exactly. OP, take note!
16. Is there some Latex code which you must follow? Or do you highlight it or something after to latex it?
17. Maybe read the forum rules?

(maths forum rules.)
18. (Original post by psanghaLFC)
Is there some Latex code which you must follow? Or do you highlight it or something after to latex it?
*Mini* guide to latex:

Enclose everything within ]latex[ ]/latex[ tags (flip brackets). Attempt to type normally. Things that need to be should be grouped with {}. ax^3+by^{3x+1} is . \frac{a}{b} is a/b, \sqrt{x} is obvious. That's enough to get by with.
19. (Original post by pyrolol)
*Mini* guide to latex:

Enclose everything within ]latex[ ]/latex[ tags (flip brackets). Attempt to type normally. Things that need to be should be grouped with {}. ax^3+by^{3x+1} is . \frac{a}{b} is a/b, \sqrt{x} is obvious. That's enough to get by with.
@pyrolol: if you want to tell someone how to use a tag, use the [noparse] tag; e.g. [noparse][latex][/noparse] will appear as [latex].

If someone else posts something in LaTeX and you'd like to know how it works, you can use the "quote" button to see what the raw LaTeX was that they used.

Most importantly:

Preview before posting. It's really easy to end up with completely unintelligible garbage because you missed out a closing } or similar.
20. (Original post by DFranklin)
@pyrolol: if you want to tell someone how to use a tag, use the [noparse] tag; e.g. [noparse][latex][/noparse] will appear as [latex].

If someone else posts something in LaTeX and you'd like to know how it works, you can use the "quote" button to see what the raw LaTeX was that they used.

Most importantly:

Preview before posting. It's really easy to end up with completely unintelligible garbage because you missed out a closing } or similar.
Nested [noparse] tags! *brain explodes*.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: July 19, 2008
Today on TSR

### 10,400 people unaware they have HIV

Have you been tested?

### University open days

• University of Roehampton
Sat, 17 Nov '18
• Edge Hill University
Faculty of Health and Social Care Undergraduate
Sat, 17 Nov '18
• Bournemouth University
Sat, 17 Nov '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE