Can you help me this problem: "Find a sequence {x_n} satisfy: $\displaystyle x_{2n+1}=3x_n +2\quad \forall n=0,1,2,\ldots$ ". Thanks to your help.
Why not just "write down" a sequence using those conditions? There are, of course, an infinite number of such sequences. That formula says nothing about even indices so we can just set them to be 0.
If we take $\displaystyle a_1= 0$, $\displaystyle a_2= 0$ (just because it is simplest), Then $\displaystyle x_3= x_{2(1)+1}= 3(0)+ 2= 2$, $\displaystyle x_4= 0$, $\displaystyle x_5= x_{2(2)+ 1}= 3(0)+ 2= 2$, $\displaystyle x_6= 0$, $\displaystyle x_7=x_{2(3)+ 1}= 3(2)+ 2= 8$, $\displaystyle x_8= 0$, [tex]x_n= x_{2(4)+ 2= 2, $\displaystyle x_{10}= 0$, etc.