Here is one of questions:
The question requires students to use mathematical induction to prove that.
I’d like to give another ways.
By looking at , we can solve just like what we do in solving second order linear differential equation. What? Urm, the characteristic equation is
With initial conditions and , we have
On solving, , thus
The simplicity of the method 1 is actually the result of iteration, let’s present it more explicitly.
– – – – – – (*)
Now apply (*) again and again,
. . . inductively . . .
– – – – – – (#)
Now we apply (#) again and again,
. . . inductively. . .
(replace by )