Your Algebra Homework Can Now Be Easier Than Ever!

Solution of Math Problem No.3

Problem: Show that a real number q is rational if and only if there are three distinct
integers, , such that forms a geometric progression.

Solution (by Richard B. Eden, Math. Graduate student, Purdue)

Suppose q is a rational number . If q = 0, we can choose . So now
suppose , not necessarily in lowest terms , where r, s ∈ Z, s ≠ 0 and r ≠ 0. We can
also assume since we can multiply r and s by the same constant.

Let . These three integers are distinct since r ≠ 0 and
s ≠ -1,-2. In this case,

really do form a geometric sequence.

Now suppose form a geometric sequence with distinct
integers. This means , which implies

If , so , we can write and for
some d ∈ Z. We then have, from the above equation ,

so d = 0 and . However, are all distinct. Therefore,
and

which is a rational number .

Also solved by :

Undergraduates: Noah Blach (Fr. Math ), Nathan Claus (Fr. Math)

Graduates: Miguel Hurtado (ECE)

Others: Brian Bradie (Christopher Newport U. VA), Hoan Duong (San Antonio College),
Elie Ghosn (Montreal, Quebec), Brian Huang (Jr. Saint Joseph's HS, IN), Gerard D.
Ko & Swami Iyer (U. Massachusetts, Boston), Steven Landy (IUPUI Physics), Sorin
Rubinstein (TAU faculty, Israel), Steve Spindler (Chicago), Kevin Ventullo (IIT, Chicago)

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of March 28th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.