Final answer:
The distributive property, a(x+y) = ax + ay, can be proven through mathematical induction by establishing a base case for two elements and then assuming it holds for k elements to prove for k+1 elements, thereby confirming its validity for any number of elements.
Step-by-step explanation:
Proving the Distributive Property using Induction
The problem at hand is to prove the distributive property of multiplication over addition, which states that a(x+y) = ax + ay. The student is asked to verify this algebraic property through mathematical induction, which is a method of proof used in mathematics to demonstrate the truth of an infinite number of cases after establishing a base case and an induction step.
Base Case
Let's start with the base case where we assume the functionality for two items. The property states that if we have two numbers x and y, then a(x+y) should be equal to ax + ay. This is known to be true for any real numbers x, y, and a.
Inductive Step
Now, for the inductive step, we assume that for some integer k the property holds, that is a(x+y+z+...+k) is equal to ax + ay + az + ... + ak. To prove the property for k+1 items, we consider a(x+y+z+...+k+(k+1)), and by using the assumed property for k items, we can express this as ax + ay + az + ... + ak + a(k+1), thereby proving the distributive property by induction.