A Question about Econometrics

#1
Suppose we estimate the following model:( y) ̂= .78t, where t is the time variable (t=1, 2, …, T). Suppose we calculate the residuals, res, and then we estimate: 〖res〗_t=β*t+e_t.
What will the estimate of β be, and why?
Suppose we propose to estimate: 〖res〗_t=β_1*t+ρ_1*〖res〗_(t-1)+e_t. What will the estimate of β be, and why?