Consider sequence $a_n$ defined by
$$a_1=2 $$
$$7a_{n+1} = a_n^2+3, n\ge2$$

  • Show that $\frac{1}{2}\lt a_n \lt 3$
  • Prove that it is increasing.
  • Find its limit as $n\to+\infty$.

I know how to prove the first part by induction. However, I don't know how to prove the second part.
And if the sequence is both bounded and increasing, which is monotonic, it is going to converge, right? But I don't think it is going to converge at all.

  • 3
    $\begingroup$ a_2=1, how is it increasing? $\endgroup$ – Joel Moreira Nov 10 '15 at 3:10
  • $\begingroup$ It is not an increasing sequence $\endgroup$ – Kushal Bhuyan Nov 10 '15 at 3:11
  • 2
    $\begingroup$ Wait, the formula is only for $n\geq2$. But then what is $a_2$? $\endgroup$ – Joel Moreira Nov 10 '15 at 3:11
  • $\begingroup$ I am not sure. That's what it said in the question. I have to ask my professor about it. $\endgroup$ – abuchay Nov 10 '15 at 3:12
  • $\begingroup$ the formula should start with $n\geq 1$ or there is a value for $a_2$ $\endgroup$ – Kushal Bhuyan Nov 10 '15 at 3:15

First, for any value of $a_n$, $a_{n+1}>3/7$. So, the sequence is bounded below by $3/7$. Second, if $a_n<3$, then $a_{n+1}<12/7<3$.

Therefore, if the initial value $a_1<3$, then $3/7<a_n<12/7<3$ for $n\ge2$.

Next, let's examine the first difference $a_{n+1}-a_n$. We have

$$\begin{align} a_{n+1}-a_n&=\frac{a_n^2+3}{7}-a_n\\\\ &=\frac{a_n^2+7a_n+3}{7}\\\\ &=\frac17 \left(a_n-\frac{7+\sqrt{37}}{2}\right)\left(a_n-\frac{7-\sqrt{37}}{2}\right)\\\\ \end{align}$$

Therefore, $a_{n+1} < a_n$ whenever $0<\frac{7-\sqrt{37}}{2}<a_n<\frac{7+\sqrt{37}}{2}<7$.

Inasmuch as $a_1=2$ $a_n$ is a monotonically decreasing sequence that is bounded below by $3/7$. Then, by the Monotone Convergence Theorem, $a_n$ converges.

Finally, letting $\lim_{n\to \infty}a_n=L$, we can write

$$\begin{align} \lim_{n\to \infty}a_{n+1}&=\lim_{n\to \infty}\frac{a_n^2+3}{7}\\\\ L&=\frac{L^2+3}{7}\\\\ L^2-7L+3&=0\\\\ L&=\bbox[5px,border:2px solid #C0A000]{\frac{7-\sqrt{37}}{2}} \end{align}$$

where we ruled out the solution $L=\frac{7+\sqrt{37}}{2}>13/2$.


It is converging and the limit is going to be the smaller solution of $x^2-7x+3=0$, which is ${7-\sqrt{37}\over2}$.

Also it is strictly decreasing. Proof:

Base case when $n=1,2$ is trivial and $a_n>a_{n+1}\implies {{a_n}^2+3\over7}>{{a_{n+1}}^2+3\over7}\implies a_{n+1}>a_{n+2}$


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