Find the sum of the first 15 terms of the geometric sequence shown below. \[ 4,8,16,32, \ldots \]

Step 1 ：We are given a geometric sequence where each term is twice the previous term. The sequence starts with 4 and the common ratio is 2. We are asked to find the sum of the first 15 terms of this sequence.

Step 2 ：The sum of the first n terms of a geometric sequence can be found 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, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 3 ：In this case, \(a = 4\), \(r = 2\), and \(n = 15\). We can substitute these values into the formula to find the sum of the first 15 terms.

Step 4 ：Substituting the values into the formula, we get \(S_{15} = 4 \times \left(1 - 2^{15}\right) / \left(1 - 2\right)\).

Step 5 ：Solving the above expression, we find that the sum of the first 15 terms of the geometric sequence is 131068.

Step 6 ：Final Answer: The sum of the first 15 terms of the geometric sequence is \(\boxed{131068}\).