Many algebra proofs are done using proof by mathematical induction. To demonstrate the power of mathematical induction, we shall prove an algebraic equation and a geometric formula with induction.

If you are not familiar with with proofs using induction, carefully study proof by mathematical induction given as a reference above. Otherwise, you could struggle with these algebra proofs below

Prove by mathematical induction that 1 + 2 + 4 + 8 + ... + 2

Show that the equation is true for n = 2. n = 2 means adding the first two terms

1 + 2 = 3 and 2

Just for fun, let's show it is true also for n = 4. n =4 means adding the first 4 terms

1 + 2 + 4 + 8 = 15 and 2

Suppose it is true for n = k

Just replace n by k

1 + 2 + 4 + 8 + ... + 2

Prove it is true for n = k + 1

You need to write down what it means for the equation to be true for n = k + 1

Here is what it means for n = k + 1:

After you replace k by k+1, you get :

1 + 2 + 4 + 8 + ... + 2

1 + 2 + 4 + 8 + ... + 2

You can now complete the proof by using the hypothesis in step # 2 and then show that

1 + 2 + 4 + 8 + ... + 2

starting with the hypothesis, 1 + 2 + 4 + 8 + ... + 2

ask yourself, " What does the next term look like? "

Since the last term now is 2

Add 2

1 + 2 + 4 + 8 + ... + 2

The trick here is to see that 2

1 + 2 + 4 + 8 + ... + 2

= 2 × 2

= 2

= 2

Show by mathematical induction that

the sum of the angles in an n-gon = ( n - 2 ) × 180

A couple of good observations before we prove it:

An n-gon is a closed figure with n sides. For example,

an n-gon with 4 sides is called a quadrilateral

an n-gon with 3 sides is called a triangle

I am now ready to show the proof.

Show that the equation is true for n = 3. Notice that n cannot be smaller than 3 since we cannot make a closed figure with just two sides or one side.

When n = 3, we get a triangle and the sum of the angles in a triangle is equal to 180

When n = 3, ( 3 - 2 ) × 180

When n = 4, you are adding one more triangle to get two triangles and the sum of the angles of the two triangles is equal to 360

When n = 4, ( 4 - 2 ) × 180

Thus, the formula is true for n = 3 and n = 4

Suppose it is true for n = k

Just replace n by k

The sum of the angles in a k-gon = ( k - 2 ) × 180

Prove it is true for n = k + 1

You need to write down what it means for the equation to be true for n = k + 1

Here is what it means for n = k + 1:

After you replace k by k+1, you get :

The sum of the angles in a (k+1)-gon = ( k +1 - 2 ) × 180

The sum of the angles in a (k+1)-gon = ( k - 1 ) × 180

You can now complete the proof by using the hypothesis in step # 2 and then show that

The sum of the angles in a (k+1)-gon = ( k - 1 ) × 180

starting with the hypothesis, the sum of the angles in a k-gon = ( k - 2 ) × 180

ask yourself, " What does the next term look like? "

Since the last term now is k-gon or a figure with k sides, the next term should be a figure with k + 1 sides after replacing k by k + 1

Now, what are we adding to both sides?

First, recall the meaning of adding one side. It means that you will be adding also one triangle

( k + 1) gon = k-gon + ( 1 side or one more triangle)

( k + 1) gon = k-gon + one more triangle

( k + 1) gon = k-gon + 180

So add 180

The sum of the angles in a k-gon + 180

The sum of the angles in a (k+1)-gon = ( k - 2 ) × 180

= 180

= 180

= 180

Other important algebra proofs:

The algebra proofs below don't use mathematical induction.

Proof of the Pythagorean theorem

A proof of the Pythagorean Theorem by president Garfield is clearly explained here.

Proof of the quadratic formula

Here we prove the quadratic formula by completing the square.