Algebra Proofs

The Three Steps of Proof by Induction

Both proofs below follow these steps.

1. The first case: show that the statement is true for the first value of n.
2. The assumption: suppose the statement is true for n = k.
3. The next case: use step 2 to prove that it is also true for n = k + 1.

Proof 1: An Algebra Equation

Prove by mathematical induction that

Step 1: Show that the equation is true for n = 1

n = 1 means taking only the first term, 1.

Both sides equal 1, so the equation is true for n = 1.

Just for fun, let’s check n = 2 and n = 4 too:

Step 2: Suppose the equation is true for n = k

Just replace n with k. This is called the hypothesis:

Step 3: Prove that it is true for n = k + 1

First, write down what it means for the equation to be true for n = k + 1. Replace k with k + 1:

Now start with the hypothesis from step 2 and ask yourself: “What does the next term look like?” The last term is 2^{k − 1}, so the next term is 2^k.

Add 2^k to both sides of the hypothesis:

The trick is to see that 2^k + 2^k = 2 × 2^k = 2¹ × 2^k = 2^{k + 1}. So the right side becomes:

This is exactly what we needed to prove, so 1 + 2 + 4 + … + 2^{n − 1} = 2ⁿ − 1 for every n ≥ 1.

Proof 2: A Geometry Formula

Show by mathematical induction that

Two observations before the proof

Observation 1: an n-gon is a closed figure with n sides. For example, an n-gon with 3 sides is a triangle, and an n-gon with 4 sides is a quadrilateral.

Observation 2: look closely at the figure above. Every time you add a side, you also add one more triangle.

Step 1: Show that the formula is true for n = 3

n cannot be smaller than 3, because you cannot make a closed figure with only one or two sides.

When n = 3, the figure is a triangle, and the angles of a triangle add up to 180°:

When n = 4, you add one more triangle, so you have two triangles, and their angles add up to 360°:

So the formula is true for n = 3 and n = 4.

Step 2: Suppose the formula is true for n = k

Just replace n with k. This is the hypothesis:

Step 3: Prove that it is true for n = k + 1

Write down what it means for the formula to be true for n = k + 1. Replace k with k + 1:

Start with the hypothesis and ask yourself: “What does the next figure look like?” The next figure has k + 1 sides.

Recall that adding one side also adds one triangle, and the angles of a triangle add up to 180°:

So add 180° to both sides of the hypothesis:

This is exactly what we needed to prove, so the sum of the angles in an n-gon is (n − 2) × 180° for every n ≥ 3.

More Proofs and Logic

Keep exploring reasoning and proof.