So I have this question where I have to find the explicit formula using iteration, and then simplify that formula and use mathematical induction to prove it. I'm really not good at this, and our teacher has a hard time explaining it to us.
$$a_k=\frac{a_{k-1}}{1+{a_{k-1}}}$$ for all integers $$k\geq1$$
$$a_1=1$$
If you compute few terms of the sequence $(a_k)$, you can easily conjecture that $a_k=1/k$. It has been given that $a_1=1$ and $a_2 =\frac{a_1}{1+a_1}=\frac{1}{2}$, $a_3=\frac{1}{3}$, $a_4=\frac{1}{4}$ and so on.
So you can claim here that terms of the sequence $(a_k)$ are given by $a_k=\frac{1}{k}$ and I will leave it for you to prove it using induction.