Here's the first one . . .
Verify . . . true!
Add to both sides:
The right side is: .
Hence, we have: .
And we have proved . . . Therefore, is true.
2) is divisible by 3 for n = 1, 2, 3,...
The n = 1 case is trivial: .
So assume the problem is true for n = k. Then we need to show that
is divisible by 3.
Let for some integer m. Then we know that .
Thus is divisible by 3.
If you know modular Mathematics:
(mod 3) then (mod 3). Thus (mod 3). Thus (mod 3).
Thus is also divisible by 3.
7) T(0) = 1 and T(n) = 2nT(n-1). Show that .
First, to get rid of the n = 0 case:
So assume this is true for n = k.
We need to show that
If the theorem is true for n = k, then . So we need to show that
Well, rearranging the LHS a bit: