To find the sum of the given series, we can observe that each term is obtained by doubling the previous term and then subtracting 1.
Let's break down the series into three parts:
Part 1: 1, 2, 4, 8, 16, ..., which is a geometric series with a common ratio of 2.
Part 2: 31, 63, 127, ..., which is a series obtained by doubling the previous term and then subtracting 1.
Part 3: The last term in the series, which is 1000.
To find the sum of Part 1, we can use the formula for the sum of a geometric series:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
In this case, the first term is 1, the common ratio is 2, and n is the number of terms in Part 1. To find n, we need to solve the equation 2^n = 1000.
Using the formula, we can find the sum of Part 1. Then, we can sum the terms in Part 2 and add the last term of Part 3 to get the final sum of the series.