Prove the binomial theorem:
Further, prove the formulas:
First, we prove the binomial theorem by induction.
Proof. For the case on the left we have,
On the right,
Hence, the formula is true for the case .
Assume then that the formula is true for some . Then we have,
Thus, if the formula is true for the case then it is true for the case . Hence, we have proved the formula for all
As an application of the binomial theorem we then prove the two formulas.
For the first, apply the binomial theorem with . Then,
For the second, apply the binomial theorem with and . Then,