I need to prove the monotony and the constraint of the sequence $(a_n)$, and also find its boundary if $a_1= 3/2$; $a_{n + 1}^2 = 3a_n - 2, n ≥ 1$

Question

I need to prove the monotony and the constraint of the sequence $a_n$, and also find its limit if $a_1 = 3/2$; $a_{n + 1}^2 = 3a_n - 2, n ≥ 1$. Should I use the method of mathematical induction here?

Answer 1

See that $a_{n+1}^2 - a_n ^2 =3(a_n -a_{n-1})$ which means, $(a_n+a_{n+1})(a_{n+1}-a_n)=3(a_n -a_{n-1})$. First show that $a_n$ is positive as $a_{n+1}^2 +2>0$ So, the monotony of the sequence depends on the relation between $a_{n+1},a_{n}$, If $a_{n+1}>a_{n} \implies a_{n} > a_{n-1}$ This continues upto $a_2 > a_1$ which is indeed true.Thus the sequence is monotonically increasing. For boundedness, let for some $k$, $a_k<2$. We have to show by mathematical induction that $a_{k+1} <2$.Indeed $a_{k+1}^2 =3a_k -2 < 3.2 -2=4 \implies a_{k+1} <2$ , Thus the sequence is bounded by $2$. Suppose $L$ be the limit of this sequence, we have:$L^2-3L+2=0 \implies L=2$