Prove $1+3+5+\cdots+(2n-1)=n^2$ By Induction

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Let's dive into the fascinating world of mathematical induction! We're going to use this powerful technique to prove that the sum of the first n odd natural numbers is equal to n squared. In simpler terms, we want to show that 1+3+5+⋯+(2n−1)=n21 + 3 + 5 + \cdots + (2n - 1) = n^2 holds true for every natural number n. This is a classic example, and understanding how to prove it will give you a solid foundation in mathematical induction.

What is Mathematical Induction?

Before we jump into the proof, let's quickly recap what mathematical induction is all about. It's a method of proving statements that hold for all natural numbers (1, 2, 3, and so on). Think of it like a line of dominoes: if you can knock over the first domino, and you can show that each domino will knock over the next one, then you know all the dominoes will fall.

Mathematical induction involves three key steps:

  1. Base Case: Show the statement is true for the smallest natural number (usually n = 1).
  2. Inductive Hypothesis: Assume the statement is true for some arbitrary natural number k. This is our assumption, the 'if' part of our domino effect.
  3. Inductive Step: Prove that if the statement is true for k, then it must also be true for the next natural number, k + 1. This is where we show that one domino knocks over the next.

If we can successfully complete these three steps, we've proven the statement for all natural numbers!

The Proof: 1+3+5+⋯+(2n−1)=n21+3+5+\cdots+(2n-1)=n^2

Now, let's apply this to our specific problem. We want to prove that the sum of the first n odd natural numbers is equal to n squared. Here's how we do it:

1. Base Case (n = 1)

First, we need to show that the statement is true for the smallest natural number, n = 1. This is the foundation of our domino effect.

When n = 1, the left side of the equation is simply the first odd natural number, which is 1. The right side is 121^2, which is also 1.

So, we have:

1=121 = 1^2

This is clearly true, so our base case holds! We've knocked over the first domino.

2. Inductive Hypothesis

Next, we assume that the statement is true for some arbitrary natural number k. This is our inductive hypothesis. It's like assuming one of the dominoes in the middle of the line will fall.

We assume that:

1+3+5+⋯+(2k−1)=k21 + 3 + 5 + \cdots + (2k - 1) = k^2

This is a crucial step. We're not proving this yet; we're simply assuming it's true for some k. This assumption will help us in the next step.

3. Inductive Step

Now comes the crucial part: we need to show that if the statement is true for k, then it must also be true for k + 1. This is where we show that one domino knocks over the next.

We want to prove that:

1+3+5+⋯+(2(k+1)−1)=(k+1)21 + 3 + 5 + \cdots + (2(k + 1) - 1) = (k + 1)^2

In other words, we want to show that the sum of the first (k + 1) odd natural numbers is equal to (k + 1) squared.

Let's start with the left side of the equation and try to manipulate it to look like the right side.

We can rewrite the left side as:

1+3+5+⋯+(2k−1)+(2(k+1)−1)1 + 3 + 5 + \cdots + (2k - 1) + (2(k + 1) - 1)

Notice that the first part of this expression, 1+3+5+⋯+(2k−1)1 + 3 + 5 + \cdots + (2k - 1), is exactly what we assumed was equal to k2k^2 in our inductive hypothesis! So, we can substitute k2k^2 for that part:

k2+(2(k+1)−1)k^2 + (2(k + 1) - 1)

Now, let's simplify the expression:

k2+(2k+2−1)k^2 + (2k + 2 - 1) k2+(2k+1)k^2 + (2k + 1)

And finally:

k2+2k+1k^2 + 2k + 1

Hey, this looks familiar! It's a perfect square trinomial, and we can factor it:

(k+1)2(k + 1)^2

This is exactly what we wanted to show! We've successfully transformed the left side of the equation into the right side, (k+1)2(k + 1)^2.

Therefore, we've proven that if the statement is true for k, then it's also true for k + 1. This completes our inductive step.

Conclusion

We've completed all three steps of mathematical induction:

  1. We showed the base case is true for n = 1.
  2. We assumed the statement is true for some arbitrary natural number k (the inductive hypothesis).
  3. We proved that if the statement is true for k, then it's also true for k + 1 (the inductive step).

By the principle of mathematical induction, we can confidently conclude that the statement 1+3+5+⋯+(2n−1)=n21 + 3 + 5 + \cdots + (2n - 1) = n^2 is true for all natural numbers n.

Isn't that neat? We've used a powerful technique to prove a mathematical truth that holds for an infinite number of cases. Mathematical induction is a fundamental tool in mathematics, and understanding it opens doors to proving many other interesting results.

Why is this useful?

This specific result, that the sum of the first n odd numbers is n², might seem like a neat mathematical curiosity, but it actually has some interesting connections to other areas. For example, it can be visualized geometrically. Imagine building squares out of unit blocks.

  • A 1x1 square requires 1 block (1² = 1).
  • To make a 2x2 square, you need 4 blocks (2² = 4). You add 3 blocks to the original 1x1 square.
  • To make a 3x3 square, you need 9 blocks (3² = 9). You add 5 blocks to the 2x2 square.

And so on. Each time you increase the size of the square, you add the next odd number of blocks. This visual representation helps to solidify the understanding of the formula.

More broadly, mathematical induction is essential in computer science for proving the correctness of algorithms and data structures, and in many other branches of mathematics. It's a fundamental concept that is well worth mastering.

Tips for Mastering Mathematical Induction

  • Practice, practice, practice! The more examples you work through, the more comfortable you'll become with the process.
  • Understand the logic behind each step. Don't just memorize the steps; make sure you understand why they work.
  • Be clear and organized in your proofs. Clearly state your base case, inductive hypothesis, and inductive step.
  • Don't be afraid to ask for help! If you're stuck, talk to your teacher, professor, or a classmate.

Further Exploration

If you're interested in learning more about mathematical induction, there are many excellent resources available online and in textbooks. You can explore other examples, such as proving formulas for the sum of the first n natural numbers, the sum of the first n squares, or inequalities.

The world of mathematical proofs is fascinating, and mathematical induction is a key to unlocking many of its secrets.

For a deeper dive into mathematical induction and its applications, you might find valuable resources at Khan Academy's Mathematical Induction Section.