With the usual notations, the following equation $S_t=u+\frac{1}{2}a(2t-1)$ is

We can derive this equation from equations of motion. So it is numerically correct.

$\begin{array}{c}{S}_t=\text{ Distance travelled in }{t}^{\text{th }}\text{ second }=\frac{\text{ Distance }}{\text{ Time }}=\left[L\right]\\ u=\text{ Velocity }=\left[L{T}^{-1}\right]\text{ and }\frac{1}{2}a(2t-1)=\left[L{T}^{-1}\right]\end{array}$

As dimensions of all terms in the given equation are not same, hence equation is not dimensionally correct.