In my book, they prove with mathematical induction propositions with successions like this:
$1 + 3 + 5 + \cdots + (2n-1) = n^2$
In all exercises. However, recently I took some exercises from a different paper and instead of these it told me to prove this:
$\forall n \in N (11 / (10^{2n+1} + 1 ))$
Or perhaps this:
$ \forall n \in N (n < 2^n) $
(I can't find how to write the natural numbers set symbol. $N$ is it there.)
And now I'm lost. This is is what I did with the first one:
Prove that the proposition works for $n=1$
$ 11 / (10^3+1) \implies 11/1001 \implies \exists x \in N(11x=1001)$ Which is true, if you take $x = 91$.
Assume $\forall n \in N (11 / (10^{2n+1} + 1 ))$ We have to prove: $\forall n \in N (11 / (10^{2n+3} + 1 ))$ We prove it:
Which I don't know how to do. Curiously enough, my book only shows exercises with successions, so I guess that this exercise can be, somehow, written as a succession? I am not sure about that. Any ideas?