1. As in the previous exercise, conjecture a formula for the value of 2n mod 7, and prove it correct 2. Suppose that we count, in binary, using an n-bit counter that goes from 0 to 2n − 1. There are 2...

1. As in the previous exercise, conjecture a formula for the value of 2n mod 7, and prove it correct

2. Suppose that we count, in binary, using an n-bit counter that goes from 0 to 2n − 1. There are 2 n different steps along the way: the initial step of 00 0, and then 2n − 1 increment steps, each of which causes at least one bit to be flipped. What is the average number of bit flips that occur per step? (Count the first step as changing all n bits.) For example, for n = 3, we have 000 → 001 → 010 → 011 → 100 → 101 → 110 → 111, which has a total of 3 + 1 + 2 + 1 + 3 + 1 + 2 + 1 = 14 bit flips. Prove your answer.