Proving n=k and n=k+1 Using Mathematical Induction
Mathematical induction is a powerful tool in mathematics that allows us to prove that a statement is true for all natural numbers (1, 2, 3, 4, and so on). The basic idea behind mathematical induction is to first prove that a statement is true for a specific case, and then use that to show that the statement is also true for the next case.
The Two-Step Process of Mathematical Induction
The process of proving a statement using mathematical induction involves two steps:
Base Case: In this step, you prove that the statement is true for a specific value of the variable, usually n=1 or n=0.
Inductive Step: In this step, you assume that the statement is true for a particular value of the variable, say n=k, and then use that assumption to prove that the statement is also true for the next value, n=k+1.
By completing these two steps, you can conclude that the statement is true for all natural numbers.
Proving n=k and n=k+1
Now, let's apply the concept of mathematical induction to the statement "n=k and n=k+1".
Base Case: Let's start by proving the statement is true for n=1.
When n=1, we have:
- n=1 (this is true by definition)
- n+1=2 (since 1+1=2)
Therefore, the statement "n=k and n=k+1" is true for n=1.
Inductive Step: Now, let's assume the statement is true for some value of n, say n=k. This means we have:
- n=k
- n+1=k+1
We need to show that the statement is also true for n=k+1.
Substituting n=k+1 into the statement, we get:
- n=k+1
- n+1=(k+1)+1=k+2
Therefore, the statement "n=k and n=k+1" is also true for n=k+1.
By completing the base case and the inductive step, we can conclude that the statement "n=k and n=k+1" is true for all natural numbers n.
This example demonstrates how the principle of mathematical induction can be used to prove that a statement holds true for all natural numbers, even when the statement involves two related expressions (n=k and n=k+1).