3. Let a, be an odd integer for cach integer n > 1. Prove that if n is odd, then the sum Ea, is odd j=1 (Hint: You want to induct on just the positive odd integers. Instead of using induction on all of n. Write these as n= 2k + 1 and induct on k.)

please do the following.  1. Prove the base case 2. Prove the inductions step, ( Start with the LHS of Sk+1 and prove that it is equal to to RHS of Sk+1) 3.Write a short sentance to justify the reason of the use of the principle of the mathematical induction.

6. The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13,... Can you see the pattern? we construct the nth number by summing the previous two numbers. Formally we can define this as: fo = 0 1 f = f = fn-2+ fn-1 Use strong induction to prove for all natural numbers n ≥ 2, that the nth Fibonacci numbers is less than 2-1., i.e., Let S(n) be fn < 2-1. Prove Vn E N22, S(n). fn < 2"-1

5. Use ordinary induction to prove that for every positive integer n, n-n is a multiple of 6. Only proofs by induction are accepted.