Hello, jpatrie!
We know that: . , where is a natural number.
Obtain this formula using the identity: .
We have: .
Let and "stack" the equations.
. .
Now add the equations ... note that most of the left side cancels out.
. .
So we have: .
Therefore: .
Then use a similar approach to obtain the formula: .
We have: .
Let
. .
Add the equations:
. . .
So we have: .
. . which simplifies to: .
Therefore: .