A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 190 responses, but the responses were declining by $10\mathrm{\%}$ each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 13 days after the magazine was published, to the nearest whole number?

Step 1 ：The problem describes a geometric sequence where the first term is 190 and the common ratio is 0.9 (since the responses are declining by 10% each day).

Step 2 ：The sum of the first n terms of a geometric sequence can be calculated using the formula: $S_n=a\ast (1-r^n)/(1-r)$ where: $S_n$ is the sum of the first n terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Step 3 ：In this case, we want to find the sum of the first 13 terms, so $n=13$. We can substitute the given values into the formula and calculate the result.

Step 4 ：Given: $a=190$, $r=0.9$, $n=13$

Step 5 ：Substitute the given values into the formula: $S_n=190\ast (1-{0.9}^{13})/(1-0.9)$

Step 6 ：Calculate the result: $S_n=1417$

Step 7 ：Final Answer: The company would get $\boxed{{\displaystyle 1417}}$ total responses over the course of the first 13 days after the magazine was published, to the nearest whole number.