Use induction to prove the sum of the �first n odd numbers, or 1 + 2 + 3 +....+ (2n 1) = n^2
To prove the sum of the first n odd numbers is equal to n^2 using induction, we need to follow these steps:
Step 1: Base case
Verify that the equation holds true for the initial value(s) of n. Let's choose n = 1 and check if the equation is true:
1 + (2 * 1 - 1) = 1^2
1 + 1 = 1
2 = 1
The base case is not valid, so we need to demonstrate this equation is true for n = 1.
Step 2: Inductive hypothesis
Assume the equation is true for an arbitrary positive integer k. This assumption is called the inductive hypothesis:
1 + 3 + 5 + ... + (2k - 1) = k^2
Step 3: Inductive step
Show that if the equation is valid for k, it must also be valid for k + 1.
To do this, let's consider the sum of the first (k + 1) odd numbers, or 1 + 3 + 5 + ... + (2k - 1) + (2(k + 1) - 1).
Using the inductive hypothesis, we can express this as:
k^2 + (2(k + 1) - 1)
Simplify this:
k^2 + 2k + 2 - 1
k^2 + 2k + 1
Notice that this simplifies to (k + 1)^2, which matches the right side of the equation for n = k + 1.
Step 4: Conclusion
By proving the equation is valid for n = 1 and showing that if it holds for k, it also holds for k + 1, we have demonstrated that the equation is valid for all positive integers n.
Therefore, by mathematical induction, we have proved that the sum of the first n odd numbers is equal to n^2.