I am given a sequence $a_1=\ln(5)$ and $a_n=\ln\left( \frac{e^{2a_{n-1}}+16}{10}\right)$, for $n\geq 2$ and I am asked to show that $a_n \geq 0$ for all $n \geq 1$.
I can do this easily with mathematical induction.
Base case: $\ln(5) \geq0$
Induction hypothesis: $a_k \geq 0$ for all $n=k$
Induction step:
$a_k \geq 0$
$2a_k \geq 0$
$e^{2a_k} \geq 1$
$e^{2a_k} + 16 \geq 17$
$\frac{e^{2a_k} + 16}{10} \geq \frac{17}{10}$
$\ln\left( \frac{e^{2_{a_k}}+16}{10}\right) \geq \ln \left( \frac{17}{10} \right) \geq 0$
$a_{k+1} \geq 0$
So because I have shown that my induction hypothesis is true for all $n=k+1$, I'm done.
HOWEVER, this is supposedly possible to do WITHOUT mathematical induction. How would I go about doing that? For context, this is an intro level calculus course that has not covered the concept of a limit yet.
Best Answer
$e^{x} >0$ for all real numbers $x$ so $a_n > ln(\frac {16} {10}) >ln 1=0$.