I am trying to prove this inequality for real $a$ and $b$ with $0\lt a \lt b \lt 1$ and integer $n \ge 0$:
$$(2n-1)a+b \lt n^2ab+1$$
I tried using induction but that approach failed. Also I tried making the smaller side greater and proving this and making the bigger side smaller and prove that, but neither of them worked. Any ideas?