To find the first five terms of a sequence when the $n^{th}$ term is provided, we must successively substitute the values of $n$ as 1, 2, 3, 4, and 5 into the given formula. The textbook context illustrates this direct method of computing terms by inserting the values of $n$ and calculating the result step-by-step.

For Part (i), the formula is linear. We substitute $n=1$ to $n=5$ into $t_n = 3n - 4$ to yield the terms -1, 2, 5, 8, and 11. This creates a sequence where each term increases by a constant difference.

For Part (ii), we use another linear formula, $t_n = 2 - 5n$. Substituting the first five natural numbers gives the sequence of terms -3, -8, -13, -18, and -23. Here, the terms decrease as $n$ increases due to the negative coefficient of $n$.

For Part (iii), the formula is quadratic: $t_n = n^2 - 2n + 3$. By substituting $n=1$, we get $1 - 2 + 3 = 2$. Similarly, substituting $n=2, 3, 4,$ and $5$ yields the values 3, 6, 11, and 18 respectively. This step-by-step substitution ensures we accurately compute each term based on its position.