CK12 - Geometry

Odd numbersform the pattern

A.Whatis the odd number?

. We can start with and add times.. Or, we noticethat eachodd

numberis less than the correspondingevennumber. The evennumberis (example

4), so the odd numberis.

B.Whatis the odd number?

. The evennumberis (example4), so the odd numberis

Example 6 – SquareNumbers

Squarenumbersform the pattern

Theseare calledsquarenumbersbecause

A.Whatis the squarenumber?

. The squarenumberis =.

B.The squarenumberis .Whatis the valueof?

. The squarenumberis =.

Conjecturesand Counterexamples

Aconjectureis an “educatedguess”that is oftenbasedon examplesin a pattern.Examplessuggesta re-
lationship,whichcan be statedas a possiblerule, or conjecture,for the pattern.

Numerousexamplesmay makeyou stronglybelievethe conjecture.However, no numberof examplescan
provethe conjecture.It is alwayspossiblethat the next examplewouldshowthat the conjecturedoesnot
work.

Example 7

Here’s an algebraicequation.

Let’s evaluatethis expressionfor somevaluesof.

Theseresultscan be put into a table.