Step 1 :Given: $m=3, t=m-1$ and $n=t+5$
Step 2 :Find the value of t: $t = m - 1 = 3 - 1 = 2$
Step 3 :Find the value of n: $n = t + 5 = 2 + 5 = 7$
Step 4 :Substitute the values of m, n, and t into the expression: $m+n^{2} / 2 t-1 = 3 + 7^{2} / 2(2) - 1$
Step 5 :Calculate the result: $3 + 49 / 4 - 1 = 3 + 12.25 - 1 = 14.25$
Step 6 :\(\boxed{14.25}\)