🔗 Beginner-Friendly Guide 'Concatenation of Consecutive Binary Numbers' - Problem 1680 (C++, Python, JavaScript)

Ever wondered how computers handle massive strings of data by simply stitching bits together? This problem challenges you to simulate that exact process by joining the binary forms of consecutive numbers into one giant value. It is a fantastic exercise in bitwise manipulation and understanding how large numbers are stored in memory. Problem Summary You're given: An integer $n$, representing a range of numbers starting from 1 up to $n$. Your goal: Concatenate the binary representations of every number in that range in order. You must then convert that resulting binary string back into a decimal integer, returning the result modulo $10^9 + 7$. Intuition: The Art of Shifting To solve this without actually creating a massive string (which would be slow and memory-heavy), we use bitwise math. Think of it like a conveyor belt. When you want to add a new number to the end of your current binary sequence, you first need to make exactly enough "room" for it. For every number $i$ from 1 to $n$,