A clock strikes one at $1\text{ o’clock}$, twice at $2\text{ o’clock}$, and so on up to twelve times at $12\text{ o’clock}$. How many times will the bell strike in the course of $2$ days?

A standard clock operates on a $12$ -hour cycle. Let's first calculate the total number of strikes in one complete $12$ -hour cycle.
The number of strikes forms an arithmetic progression: $1,2,3,\dots ,12$.

$$\text{Strikes in }12\text{ hours}=1+2+3+\cdots +12$$

Using the sum formula for the first $n$ natural numbers, $S_n=\frac{n(n+1)}{2}$:

$$\text{Strikes in }12\text{ hours}=\frac{12\times (12+1)}{2}=6\times 13=78$$

A single day has $24$ hours, which consists of two $12$ -hour cycles. Therefore, $2$ days contain four $12$ -hour cycles:

$$\text{Total strikes in 2 days}=78\times 4=312$$