Find the value of the following expression and round to the nearest integer: \[ \sum_{n=2}^{50} 600(1.05)^{n-2} \]

Step 1 ：We are given the expression \(\sum_{n=2}^{50} 600(1.05)^{n-2}\) and asked to find its value, rounded to the nearest integer.

Step 2 ：This is a geometric series with first term 600, common ratio 1.05, and 49 terms.

Step 3 ：The sum of a geometric series can be calculated using the formula: \[S = a \frac{1 - r^n}{1 - r}\] where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

Step 4 ：In this case, a = 600, r = 1.05, and n = 49. We can substitute these values into the formula to find the sum of the series.

Step 5 ：Doing so, we find that S = 119055.99755147073.

Step 6 ：Rounding this to the nearest integer, we get S = 119056.

Step 7 ：Thus, the value of the expression, rounded to the nearest integer, is \(\boxed{119056}\).