Use strong induction to show that every positive integer n can be written as a sum of distinct powers of two, that is, as a sum of a subset of the integers 20 = 1, 21 = 2, 22 = 4, and so on. Let P(n) be the proposition that the positive integer n can be written as a sum of distinct powers of 2. Click and drag the given steps (in the right) to the corresponding step names (in the left) to show that if P() is true for all j≤ k, then P(k+ 1) is also true. First prove the above statement when k + 1 is odd and then prove when k + 1 is even. Step 1 Step 2 Step 3 Step 4 If k + 1 is even, then (k+ 1)/2 is a positive integer, so by the inductive hypothesis (k+ 1)/2 can be written as a sum of distinct powers of 2. Increasing each exponent by 1 doubles the value and gives us the desired sum for k + 1. If k+ 1 is odd, then (k+ 1)/2 is a positive integer, so by the inductive hypothesis (k+ 1)/2 can be written as a sum of distinct powers of 2. If k + 1 is even, then k is even, so 2° was not part of the sum for k Therefore the sum for k+ 1 is the same as the sum for k with the extra term 2º added. If k+ 1 is odd, then k is even, so 2° was not part of the sum for k.

Let k be a positive integer. Use induction to prove that (k – 1)"+1+k2n-1 is divisible by k2 – k+1 for every positive integer n.

Use proof by contraposition to prove this.   If n is an integer and 3n is not a multiple of six, then n is odd.

Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms in the aggregate being combined by the signs + and - appropriately chosen.

. Use the principle of ordinary mathematical induction to prove the well-ordering principlefor the integers.

Use mathematical induction to prove that for each positive integer n

Prove parts a and d

Use induction to prove that for every integer n  ≥ 0, 5n+2+ 62n+1 is divisible by 31.

Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0) = S F Recursive step: If (a, b) = S, then (a, b + 1) = S, (a + 1, b + 1) = S, and (a + 2, b + 1) = S. List the elements of S produced by the first four applications of the recursive definition. Enter your answers in the form (a₁, b₁), (a2, b2),..., (an, bn), in order of increasing a, without any spaces. The first application of the recursive step adds (Click to select) ✓to S. The second application of the recursive step adds (Click to select) The third application of the recursive step adds (Click to select) The fourth application of the recursive step adds (Click to select) to S. ✓to S. ✓to S.

Given an integer N with an odd number of digits. Let M be the integer formed by reversing the order of the digits.(for example, if N = 123, then the integer formed by reversing the order of the digits of N is M = 321).Show that (N) power 11- (M) power 11 is divisible by 11.