Mathematical induction facts for kids

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.

Mathematical induction is a special way to prove that a mathematical idea is true. It helps us show something is true for all natural numbers. Natural numbers are all the positive whole numbers like 1, 2, 3, and so on.

Think of it like a line of dominoes:

• First, you make sure the very first domino falls over.
• Then, you show that if any domino falls, the next one in line will also fall.
• If both of these things are true, then you know *all* the dominoes will fall!

In math, we use these steps:

• Step 1: The Starting Point (Base Case)

* You show that the idea is true for the very first number, usually 1 (or sometimes 0, or another starting number). This is like pushing the first domino.

• Step 2: The Chain Reaction (Inductive Step)

* You pretend the idea is true for *any* natural number, let's call it n. * Then, you prove that if it's true for n, it *must* also be true for the next number, which is n+1. This is like showing that if one domino falls, it knocks over the next one.

Because the first step is true, and because each step leads to the next, the idea becomes true for 1, then for 2 (because 1 leads to 2), then for 3 (because 2 leads to 3), and so on, forever!

Proving a Pattern with Induction

Let's look at an example to see how mathematical induction works.

Imagine you want to prove that if you add up all the numbers from 1 to n, the answer is always the same as `n` multiplied by `(n+1)`, then divided by 2. So, `1 + 2 + 3 + ... + n = (n * (n+1)) / 2`

Let's prove this using induction:

Step 1: Check the First Case

We need to see if the idea is true for n = 1. If n = 1, the sum is just 1. Using the formula: `(1 * (1+1)) / 2 = (1 * 2) / 2 = 2 / 2 = 1`. Since 1 = 1, the idea is true for n = 1. Our first domino falls!

Step 2: Assume and Prove the Next Case

Now, we pretend the idea is true for some number n. This is our "assumption." So, we assume that `1 + 2 + 3 + ... + n = (n * (n+1)) / 2` is true.

Next, we need to prove that it's also true for the *next* number, which is n + 1. This means we want to show that: `1 + 2 + 3 + ... + n + (n+1) = ((n+1) * ((n+1)+1)) / 2` `1 + 2 + 3 + ... + n + (n+1) = ((n+1) * (n+2)) / 2`

Let's start with the left side of this new equation: `1 + 2 + 3 + ... + n + (n+1)`

We know from our assumption that `1 + 2 + 3 + ... + n` is equal to `(n * (n+1)) / 2`. So, we can replace that part: `(n * (n+1)) / 2 + (n+1)`

Now, let's do some algebra to simplify this expression. We can find a common denominator: `(n * (n+1)) / 2 + (2 * (n+1)) / 2`

Now, combine them: `(n * (n+1) + 2 * (n+1)) / 2`

Notice that `(n+1)` is in both parts of the top. We can factor it out: `((n+1) * (n + 2)) / 2`

This is exactly what we wanted to prove! So, if the idea is true for n, it's also true for n + 1. Our dominoes keep falling!

Since both steps are true, we have proven that `1 + 2 + 3 + ... + n = (n * (n+1)) / 2` is true for all natural numbers.

Other Ways to Start Induction

Mathematical induction doesn't always have to start with 1. You can start with any number. For example, you could prove something is true for all natural numbers starting from 0, or 3, or any other number.

Here's an example where we start with 3: The total degrees inside a polygon (a shape with straight sides) with n sides is `(n-2) * 180` degrees.

Step 1: Check the First Case (Starting with 3)

The smallest polygon is a triangle, which has 3 sides. So, our starting n is 3. For a triangle (n=3), the formula says: `(3-2) * 180 = 1 * 180 = 180` degrees. We know a triangle's angles add up to 180 degrees, so this is true.

Step 2: Assume and Prove the Next Case

Assume that the sum of angles in an n-sided polygon is `(n-2) * 180` degrees. Now, imagine we have an n-sided polygon. If we add one more side to it to make an (n+1)-sided polygon, we can do this by adding a triangle to one of its sides. Adding a triangle increases the number of sides by 1, and it adds 180 degrees to the total angle sum. So, the new sum of angles would be: `(n-2) * 180 + 180`

We can factor out 180: `180 * ((n-2) + 1)` `180 * (n-1)`

This can also be written as: `((n+1)-2) * 180`

This matches the formula for an (n+1)-sided polygon! So, the proof works.

Defining Things with Induction

The same idea of induction can be used to *define* things, not just prove them.

For example, we can define what an "nth degree cousin" is:

• A 1st degree cousin is the child of your parent's sibling (your aunt or uncle).
• An n+1st degree cousin is the child of your parent's nth degree cousin.

This way, you can figure out what a 2nd degree cousin is (child of your parent's 1st degree cousin), and a 3rd degree cousin, and so on, all based on the first definition.

See also

In Spanish: Inducción matemática para niños