example of contractive sequence

Define the sequenceMathworldPlanetmatha1,a2,a3,  by

a1:=1,an+1:=5-2an(n=1,2,3,). (1)

We see by inductionMathworldPlanetmath that the radicand in (1) cannot become negative; in fact we justify that

1an3 (2)

for every n:  It’s clear when n=1. If it is true for an an, it implies that 1<5-2an3, i.e. 1<an+13.

As for the convergence of the sequence, which is not monotonic, one could think to show that it is a Cauchy sequence. Unfortunately, it is almost impossible to directly express and estimate the needed absolute valueMathworldPlanetmathPlanetmathPlanetmath of am-an. Fortunately, the recursive definition (1) allows quite easily to estimate |an-an+1|.  Then it turns out that it’s a question of a contractive sequence, whence it is by the parent entry (http://planetmath.org/ContractiveSequence) a Cauchy sequence.

We form the differencePlanetmathPlanetmath

an-an+1 =(5-2an-1-5-2an)(5-2an-1+5-2an)5-2an-1+5-2an

where n>1.  Thus we can estimate its absolute value, by using (2):


Since 15-23<1, our sequence (1) is contractive, consequently Cauchy.  Therefore it convergesPlanetmathPlanetmath to a limit A.

We have


From the quadratic equationA2+2A-5=0  we get the positive root A=6-1. I.e.,

limnan=6-1. (3)