Questions by kobe08 - Page 37
Click and drag the given steps (in the right) to the corresponding step names (in the left) to show the inductive step to prove that P(n) is true. Step 1 If k + 1 is odd, then k is even, so 2 hat 0 was not part of the sum for k. Step 2 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. Step 3 Increasing each exponent by 1 doubles the value and gives us the desired sum for k + 1. Step 4 Therefore the sum for k + 1 is the same as the sum for k with the extra term 2 hat 0 added. If k + 1 is odd, then k is even, so 2 hat 0 was not part of the sum for k. Increasing each exponent by 1 doubles the value and gives us the desired sum for k + 1. 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. 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 hat 0 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 2 hat 0 added.