- Prove the following using only the rules of natural deduction for predicate logic with equality, the Peano Axioms, and the recursive definitions of the symbols given in the lecture notes. You can Your proof should be written in the semiformal proof style using any of the shortcuts we went over. You can use any of the theorems about natural numbers that appear at the end of section 7.5 in the lecture notes. All quantified variables represent natural numbers. You cannot use any negative signs or subtraction in your proofs because we have not yet defined those. Feel free to ask an instructor for help if you get stuck.
**1. Row sum in Pascal’s triangle.**$$ \sum_{k=0}^n {n \choose k}=2^n$$**2. Factorial vs. Exponential Growth.**For all natural numbers $n\geq 4$, $$2^n\lt n!$$**3. A Fibonnaci binomial theorem?**Suppose $n$ is a natural number. Then $$\sum_{k=0}^n \binom{n}{k}\cdot F_k = F_{2n}$$