Find the sum of the series \(1 + 2 + 4 + 8 + 16 + ... + 1024\)

Step 1 ：This is a geometric series with first term \(a = 1\) and common ratio \(r = 2\). The sum of the first \(n\) terms of a geometric series can be found using the formula \(S_n = \frac{a(r^n-1)}{r-1}\). We need to find the value of \(n\) such that \(2^n = 1024\).

Step 2 ：Solving the equation \(2^n = 1024\), we get \(n = 10\).

Step 3 ：Substituting \(a = 1\), \(r = 2\), and \(n = 10\) into the formula, we get \(S_{10} = \frac{1(2^{10}-1)}{2-1} = 1024 - 1 = 1023\).