# Unsure how to do this problem with iteration and mathematical induction

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 $$k\geq1$$

$$a_1=1$$

• The (education) tag is meant to be used for questions about pedagogy and the action of teaching, for example "how many tests or quizzes should be given in a semester." It is not to be used just because you encountered your question in a school setting. The tag (proof-verification) is meant to be used when you have a complete attempt at a proof and you want to check that your attempt is correct. It is not meant to be used when you are asked to prove something and your attempt is incomplete or nonexistent. – JMoravitz Oct 15 '18 at 2:39

## 1 Answer

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.