Present a relationship with the domain some subset of the integers that is to be proven via Mathematical Induction.
Present a relationship with the domain some subset of the integers that is to be proven via Mathematical Induction. A model response might look like this:
n(n+2) is divisible by 4 for all EVEN positive Integers.
Prove your relationship with the domain some subset of the integers via Mathematical
Induction. Note you must detail each of the four steps and show all of your work.
o A model response might look like this:
Step 1: Show true for first element of the Domain (i.e., n = 2)
2(2 + 2) = 2 * 4 = 8 is divisible by 4
Step 2: Assume true for arbitrary even integer:
Assume k(k+2) is divisible by 4 for any positive even integer, k
(note: divisible by 4 means can be written as 4* some Integer)
Assume: k(k+2) = 4p (where p is an integer)
Step 3: Show true for next element of the Domain (i.e., n = k + 2)
Show (k+2)(k+4) is divisible by 4
That is: show (k+2)(k+4) = 4q, for some integer, q
NOTE: show k2 + 6k + 8 = 4q
k(k+2) = 4p by the Induction Hypothesis
k2 + 2k = 4p
k2 + 2k + 4k + 8= 4p + 4k + 8
(k+2)(k+4) = 4(p + k + 2)
Step 4: Therefore, by the Principle of Mathematical Induction, n(n+2) is divisible by 4 for all
positive, even integers.
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