Intuition can be one way!

Show that the numbers $\sqrt 2, \sqrt 3, \sqrt 5$ can never be the terms (not necessarily consecutive) of a single Arithmetic Progression.

Generalized Form: Show that the numbers $\sqrt p, \sqrt q, \sqrt r$ can never be the terms (not necessarily consecutive) of a single Arithmetic Progression, where $p,q,r$ are three distinct prime numbers.

#Algebra #ArithmeticProgression(AP) #Proofs #JEE #IITJEE

Easy Math Editor

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Comments

Suppose $a_{i} = \sqrt{2}, a_{j} = \sqrt{3}, a_{k} =\sqrt{5}$ are terms in an arithmetic sequence $\{a_{n}\}$ with common difference $d,$ where $i,j,k$ are not necessarily consecutive. Then

$\sqrt{2} = a_{1} + (i - 1)d, \sqrt{3} = a_{1} + (j - 1)d$ and $a_{k} = a_{1} + (k - 1)d.$

We can then subtract these equations from one another successively to find that

$\sqrt{3} - \sqrt{2} = (j - i)d$ and $\sqrt{5} - \sqrt{3} = (k - j)d$

$\Longrightarrow \dfrac{k - j}{j - i} = \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{3} - \sqrt{2}} = \sqrt{15} + \sqrt{10} - \sqrt{6} - 3.$

But then the LHS of this last equation is rational and the RHS irrational, and so our initial assumption that $\sqrt{2}, \sqrt{3}, \sqrt{5}$ are terms of the same AP is false.

For the generalized form with primes $p,q,r$ the RHS of the last equation would become

$\dfrac{\sqrt{r} - \sqrt{q}}{\sqrt{q} - \sqrt{p}} = \dfrac{1}{q - p}(\sqrt{rq} + \sqrt{rp} - \sqrt{pq} - q),$

which is, as before, irrational, (since none of $rq, rp, pq$ are perfect squares, nor do the roots cancel).

(Note that they cannot be terms of the same geometric progression either.)

Brian Charlesworth - 5 years, 9 months ago

Greetings, Brian! Just to play the devil's advocate: How do you know that $\sqrt{15}+\sqrt{10}-\sqrt{6}$ isn't rational?

Otto Bretscher - 5 years, 9 months ago

The most rational reasoning is to use the standard method of rationalising and finding a contradiction. This should help

Sualeh Asif - 5 years, 9 months ago

Greetings, Otto! Yes, I was afraid that someone was going to ask that. I was just about to write up a proof, but I notice now that the approach I was going to use is the same as in Sualeh's link, so I think that reference will suffice. :)

Brian Charlesworth - 5 years, 9 months ago

@Brian Charlesworth – Thank you both!

Otto Bretscher - 5 years, 9 months ago

Math	Appears 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}`	$\boxed{123}$