The binomial theorem states that ∀x,y∈ℝ and ∀n,k∈ℕ, such as k≤n, we have:
- Have a knowledge of binomial coefficient.
- Have a knowledge of the principle of induction.
- Know that: x0=1.
In order to show that the binomial theorem is true for n∈ℕ. We will first prove it is true for n=0, then for n∈ℕ∗ by induction.
Proof for n=0
We then deduce that the binomial theorem is true for n=0.
Because we know that the binomial theorem is true for n=0 and for n=n+1, we deduce by induction that it is true for n∈ℕ.