Discrete Mathematics

Solve the following problems. Justify your answers. Solutions without justification will  not receive full credit.
1. How many binary strings of length n do not contain 10?
2. Find a recurrence relation for the number of length-n ternary strings (strings using
values 0, 1, 2) without two consecutive 0s. What are the initial values?
3. In how many ways can you cover a 2 × n chessboard by 2 × 1 dominoes (placed
horizontally or vertically)? Find a recurrence relation and initial values.
4. How many subsets are there of the set {1, 2, . . . , n} that DO NOT contain three consecutive integers? Find a recurrence relation and initial values.
5. Which is larger 2100 or F100?
6. Use induction to prove
F2 + F4 + F6 + · · · + F2n = F2n+1 − 1.
7. Prove (without using induction)
F2 n+1 − F2n = Fn−1Fn+2