$$() = (−1)+\frac{−1}{^2}⋅\sum_{k=1}^{n-1}t(k)$$
Use induction to prove that $()≤2$ for all $≥1$.
I have the base case.
I got \begin{align} (m+1) & = m+\frac{m}{(m+1)^2}⋅\sum_{k=1}^{m}t(k) \\ & \le m+\frac{m}{(m+1)^2}⋅\sum_{k=1}^{m}2k \\ & = m+\frac{m}{(m+1)^2}⋅m(m+1) \\ & = m+\frac{m^2}{(m+1)} \\ & \le 2m+2 \end{align}
Does that seem right?