ok, so first of all, you had different variables, everything must be in one variable. you cant make a statement about x and then prove something about n when we don't know anything about n. i changed everything to x. Here's how i would do them

Let P(x): "If

, then

"

We have for the base case:

P(1):

, so P(1) is true

Assume P(k) is true for some k

1. We show that P(k + 1) is true

Since P(k) is true, we have:

It follows that:

, since we add 3 to the left and 2 to the right

But we can factor this so that:

which is P(k + 1)

Thus the inductive proof is complete and P(x) holds true for all x

1

This one is pretty much the same as the last.

Let P(n): "If

, then

"

So P(1):

. So P(1) is true

Assume P(k) true for some k

1, we show that P(k + 1) is true

Since P(k) is true, we have:

It follows that:

Factoring we get:

, which is P(k + 1)

Thus the inductive proof is complete, and P(n) holds for all n

1