Find the sum of the first 14 terms of the geometric sequence shown below. \[ -\frac{5}{2}, 5,-10,20, \ldots \]

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

Step 2 ：From the given sequence, we can see that the first term \(a = -\frac{5}{2}\) and the common ratio \(r = -2\). We are asked to find the sum of the first 14 terms, so \(n = 14\).

Step 3 ：Let's plug these values into the formula and calculate the sum.

Step 4 ：\(a = -2.5\)

Step 5 ：\(r = -2\)

Step 6 ：\(n = 14\)

Step 7 ：\(S_n = 13652.5\)

Step 8 ：Final Answer: The sum of the first 14 terms of the geometric sequence is \(\boxed{13652.5}\).