I'm independently working through Ian Millington's book Game Physics Engine Development. Since I'm doing this in my spare time and not for school, I contacted the author who graciously provided me with answers for the exercises at the end of each chapter so I could check my work. However, exercise 3.4a is giving me trouble.
A particle beings at $\begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix}$ and is moving with velocity $\begin{bmatrix} 1 \\ -1 \\ 2 \\ \end{bmatrix}$ per second, and acceleration $\begin{bmatrix} 0 \\ 1 \\ -1 \\ \end{bmatrix}$ per second per second.
Part a of the question asks "Use equation 2.9 to calculate what its position will be after 5 s."
The equation being referred to is $p' = p + \dot pt + \ddot p \frac{t^2}{2}$ where $\dot p$ is the velocity of the object at the start of the time interval, and $\ddot p$ is the constant acceleration over the entire time.
In the author's solutions document, he lists this solution for part a: $$ \begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} + \begin{bmatrix} 1 \\ -1 \\ 2 \\ \end{bmatrix} \cdot 5 + \begin{bmatrix} 0 \\ 1 \\ -1 \\ \end{bmatrix} \cdot 25 = \begin{bmatrix} 6 \\ 22 \\ -12 \\ \end{bmatrix} $$
However, in my reading of the equation, I don't understand where the $25$ is coming from in the part of the equation listed as $\frac{t^2}{2}$. If $t = 5$, then shouldn't that part be $12.5$?
I have contacted the author for clarification, but I haven't received a response.
It appears to be a mistake with the author's solution as the $25$ should be, as you state, $12.5$, i.e., $\frac{t^2}{2}$ as stated in the formula, instead of $25$.
Note you can also confirm the equation you stated being used by doing the appropriate integrations, or checking online such as equation ($2$) in the Constant translational acceleration in a straight line section of Wikipedia's "Equations of motion" article, i.e.,
$$r = r_0 + v_0 t + \frac{1}{2}at^2 \tag{1}\label{eq1A}$$
where $t$ is the time, $r$ is the final position, $r_0$ is the initial position, $v_0$ is the initial velocity and $a$ is the constant acceleration.