A sequence of positive real numbers is defined by
•a0 = 1,
•an+2 = 2an − an+1, for n = 0, 1, 2, ... .

Find a2005.

Question

A sequence of positive real numbers is defined by
•a0 = 1,
•an+2 = 2an − an+1, for n = 0, 1, 2, ... .

Find a2005.

jamesj · Answer

you need to give us a1 as well.

zarkon · Answer

because of  "sequence of positive real numbers"

 $$a_1$$  will be forced to be 1

anonymous · Answer

I believe the 2005th term must still be 1.

Looking at the equation and creating the characteristic equation we obtain:$$a_{n+2}=-a_{n+1}+2a_n\iff a_{n+2}+a_{n+1}-2a_n=0$$$$\ \iff \lambda^2+\lambda-2=0\iff \lambda_1=-2,\lambda_2=1$$

This tells us a closed form for the sequence is in the form:$$a_n=c_1(-2)^n +c_2(1)^n=c_1(-2)^n+c_2$$where the constants depend on initial conditions.  However, because this sequence is make of nothing but positive numbers, we already know that:$$c_1=0$$, therefore we obtain:$$a_n=c_2\iff a_0=c_2\iff 1= c_2 \iff a_n=1 $$ for all n.