<s>[INST] Prove By Induction $U_n=2^n+1.$ Given that $U_1=3,U_2=5,$ and $U_{n+2}-3U_{n+1}+2U_n=0.$
Show that $U_n=2^n+1.$ 
I'm stuck at showing that if $P(n+1)$ is true if $P(n)$ is true.
 [/INST] It's hard using induction. The characteristic equation is $$z^2-3z+2=0$$which has two roots $$z_1=1\\z_2=2$$then the equation of $U_n$ is $$U_n=a\cdot2^n+b\cdot1^n$$which by substituting $U_1$ and $U_2$ yields the same result.
alternative way (induction)
Consider $w_n=u_{n+1}-u_{n}$ therefore$$w_{n+1}=2w_n$$with $w_1=2$ which leads to $$w_n=2^n$$also $$u_{n+1}=u_n+2^n$$ or $$u_{n+1}=u_1+2+2^2+2^3+2^4+\dots +2^n=2+1+2+2^2+2^3+2^4+\dots +2^n=2^{n+1}-1+2=2^{n+1}+1$$or alternatively$$u_n=2^n+1$$
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