# Mathematical Induction

Mathematical Induction is a special way of proving things. It has only 2 steps:

- Step 1. Show it is true for the
**first one** - Step 2. Show that if
**any one**is true then the**next one**is true

Then **all** are true

Have you heard of the "Domino Effect"?

- Step 1. The
**first**domino falls - Step 2. When
**any**domino falls, the**next**domino falls

So ... **all dominos will fall!**

That is how Mathematical Induction works.

In the world of numbers we say:

- Step 1. Show it is true for
**n=1** - Step 2. Show that if
**n=k**is true then**n=k+1**is also true

## How to Do it

Step 1 is usually easy, we just have to prove it is true for **n=1**

Step 2 is best done this way:

**Assume**it is true for**n=k****Prove**it is true for**n=k+1**(we can use the**n=k**case as a**fact**.)

Note: Step 2 can often be * tricky*, we may need to use imaginative tricks to make it work!

Like in this example:

### Example: is 3^{n}−1 a multiple of 2?

**Is that true? Let us find out.**

**1.** Show it is true for **n=1**

**3 ^{1}−1 = 3−1 = 2 **

Yes 2 is a multiple of 2. That was easy.

3^{1}−1 is true

**2.** Assume it is true for **n=k**

3^{k}−1 is true

(Hang on! How do we know that?
We don't!

It is an **assumption** ... that we treat

**as a fact** for the rest of this example)

Now, prove that **3 ^{k+1}−1** is a multiple of 2

**3 ^{k+1}** is also

**3×3**

^{k}And then split **3×** into **2×** and **1×**

And each of these are multiples of 2

Because:

**2×3**is a multiple of 2 (we are multiplying by 2)^{k}**3**(we said that in the assumption above)^{k}−1 is true

So:

3^{k+1}−1 is true

DONE!

Did you see how we used the **3 ^{k}−1** case as being

**true**, even though we had not proved it? That is OK, because we are relying on the

**Domino Effect**...

... we are asking **if** any domino falls will the **next one** fall?

So we take it as a fact (temporarily) that the "**n=k**" domino falls (i.e. **3 ^{k}−1** is true), and see if that means the "

**n=k+1**" domino will also fall.

## Tricks

I said before that we often need to use imaginative tricks.

A common trick is to rewrite the **n=k+1** case into 2 parts:

- one part being the
**n=k**case (which is assumed to be true) - the other part can then be checked to see if it is also true

We did that in the example above, and here is another one:

### Example: Adding up Odd Numbers

1 + 3 + 5 + ... + (2n−1) = n^{2}

**1.** Show it is true for **n=1**

1 = 1^{2} is True

**2.** Assume it is true for **n=k**

1 + 3 + 5 + ... + (2k−1) = k^{2} is True

(An assumption!)

Now, prove it is true for "k+1"

1 + 3 + 5 + ... + (2k−1) + (2(k+1)−1) = (k+1)^{2} **?**

We know that **1 + 3 + 5 + ... + (2k−1) = k ^{2}** (the assumption above), so we can do a replacement for all but the last term:

**k ^{2}** + (2(k+1)−1) = (k+1)

^{2}

Now expand all terms:

k^{2} + 2k + 2 − 1 = k^{2} + 2k+1

And simplify:

k^{2} + 2k + 1 = k^{2} + 2k + 1

**They are the same! So it is true.**

So:

1 + 3 + 5 + ... + (2(k+1)−1) = (k+1)^{2} is True

DONE!

## Your Turn

Now, here are two more examples **for you to practice** on.

Please try them first yourself, then look at our solution below.

### Example: Triangular Numbers

Triangular numbers are numbers that can make a triangular dot pattern.

Prove that the **n-th** triangular number is:

T_{n} = n(n+1)/2

### Example: Adding up Cube Numbers

Cube numbers are the cubes of the Natural Numbers

Prove that:

1^{3} + 2^{3} + 3^{3} + ... + n^{3} = ¼n^{2}(n + 1)^{2}

. . . . . . . . . . . . . . . . . .

Please don't read the solutions until you have tried the questions yourself, these are the only questions on this page for you to practice on!

### Example: Triangular Numbers

Prove that the **n-th** triangular number is:

T_{n} = n(n+1)/2

**1.** Show it is true for **n=1**

T_{1} = 1 × (1+1) / 2 = 1 is True

**2.** Assume it is true for **n=k**

T_{k} = k(k+1)/2 is True (An assumption!)

Now, prove it is true for "k+1"

T_{k+1} = (k+1)(k+2)/2 ?

We know that T_{k} = k(k+1)/2 (the assumption above)

T_{k+1} has an extra row of (k + 1) dots

So, T_{k+1} = T_{k} + (k + 1)

(k+1)(k+2)/2 = k(k+1) / 2 + (k+1)

Multiply all terms by 2:

(k + 1)(k + 2) = k(k + 1) + 2(k + 1)

(k + 1)(k + 2) = (k + 2)(k + 1)

They are the same! So it is **true**.

So:

T_{k+1} = (k+1)(k+2)/2 is True

DONE!

### Example: Adding up Cube Numbers

Prove that:

1^{3} + 2^{3} + 3^{3} + ... + n^{3} = ¼n^{2}(n + 1)^{2}

**1.** Show it is true for **n=1**

1^{3} = ¼ × 1^{2} × 2^{2} is True

**2.** Assume it is true for **n=k**

1^{3} + 2^{3} + 3^{3} + ... + k^{3} = ¼k^{2}(k + 1)^{2} is True (An assumption!)

Now, prove it is true for "k+1"

1^{3} + 2^{3} + 3^{3} + ... + (k + 1)^{3} = ¼(k + 1)^{2}(k + 2)^{2} ?

We know that 1^{3} + 2^{3} + 3^{3} + ... + k^{3} = ¼k^{2}(k + 1)^{2} (the assumption above), so we can do a replacement for all but the last term:

¼k^{2}(k + 1)^{2} + (k + 1)^{3} = ¼(k + 1)^{2}(k + 2)^{2}

Multiply all terms by 4:

k^{2}(k + 1)^{2} + 4(k + 1)^{3} = (k + 1)^{2}(k + 2)^{2}

All terms have a common factor (k + 1)^{2}, so it can be canceled:

k^{2} + 4(k + 1) = (k + 2)^{2}

And simplify:

k^{2} + 4k + 4 = k^{2} + 4k + 4

They are the same! So it is true.

So:

1^{3} + 2^{3} + 3^{3} + ... + (k + 1)^{3} = ¼(k + 1)^{2}(k + 2)^{2} is True

DONE!