Prove the following statements using mathematical induction. 1. Prove that 1 + 2 + 3 + ... + n = n(n+1)/2 for every positive integer n. a) What do you need to prove in the basis step? b) What do you need to prove in the inductive step? c) Complete the inductive step 2. Prove that 1 + 3 + 5 + ... + (2n-1) = n^2 for every positive integer n. a) What do you need to prove in the basis step? b) What do you need to prove in the inductive step? c) Complete the inductive step 3. Prove that 2^n > n^2 for every positive n that is greater than 4. 4. Prove that n^5 - n is divisible by 5 for every positive integer n. 5. Prove that 1*2 + 2*3 + 3*4 + ... + n*(n+1) = (n)(n+1)(n+2)/3 for every positive integer n. 6. a) Find a formula for 1/2 + 1/4 + 1/8 + ... + 1/(2^n) b) Prove the formula you conjectured in part (a). 7. Consider the sequence: 1 + 2 + 4 + 8 + 16 + ... What is the sum of the first n elements? Prove your answer using mathematical induction.