I am trying to solve the following problem using strong induction, the problem is the following:
For any positive integer $n$, let $T_n$ be the number $1$ if $n<4$ and the number $T_{n − 1} +T_{n − 2} +T_{n − 3}$ if $n\ge 4$.
We have $T_1 =1, T_2 =1, T_3 =1, T_4 =T_3 +T_2+T_1=1+1+1=3$, $T_5 =T_4 +T_3 +T_2 =3+1+1=5$, etc.
Prove that: $\forall n \in \Bbb{Z}^+$, $T_n <2^n$ $\;$
Sadly, I don't even know where to start this question!