For part of a homework assignment, I am being asked to prove the following:
Let $a_1=2$ and $a_n=1+a_1a_2a_3\dots a_{n-1}$ for $n\ge2$. Prove that $a_n=a_{n-1}^2-a_{n-1}+1$ for all $n\ge2$.
I am stuck. Any ideas? The homework assignment is on induction, so I am inclined to believe that it can be proved using induction, but my attempts so far have all been unsuccessful. Maybe it's not true?
Easiest, I think, to simply manipulate the recursion.
We have $$a_{n-1}-1=a_1a_2\cdots a_{n-2}$$ $$a_n-1=a_1a_2\cdots a_{n-2}a_{n-1}$$
Which implies that $$a_n-1=(a_{n-1}-1)\times a_{n-1}=a_{n-1}^2-a_{n-1}$$
and we are done.