Given a sequence $\{a_n\}$, with $n\geq1$, where   $$a_{1}=4,$$   and    $$a_{n+1}=\sqrt{a_{n} +20}.$$   Prove via induction that, for all $n \geq 1$,     $$a_{n+1}&gt;a_{n}.$$    Apparent Convergence   The sequence appears to be increasing, and possibly bounded at 5. How may I show convergence, and find its limit.   Regarding Notations   Additionally are there any beginners guide on using appropriate notations?  Thank you for your assistance.
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
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