here's the first. i want you to try the third question on your own, and post your solution.

this is how math. induction works.

we begin with some statement that depends on a set of integers, usually the whole set of natural numbers. to prove the statement is true by math. induction, we do the following procedure.

we prove that the statement is true for the first integer it is claimed to be true for, usually this is 1. so we find P(1) and show that the statement is true if we replace n with 1. if it's true, we are in good shape. this step is called

**the basis**
next we move on to the

**inductive step** or

**inductive hypothesis**. in this step, we assume that the statement is true for some integer k which is greater than or equal to the base integer. and then we prove that if it's true for this term, then it will be true for the next term, that is, the k + 1 term.

so iduction in a nutshell:

given a statement P(n) for n

1

- prove P(1) is true

- assume P(k) is true for some k

1, and derive that P(k + 1) is true

so here is the first:

Let

for all integers

Then

, which is true.

So

is true

Assume

is true for some

, we show that

is true.

So we have:

add the

term to both sides, we get:

=

.................................................. .................................. .................................................. .................................. .................................................. .................................. .................................................. .................................. .................................................. .................................. .................................................. .................................. .................................................. ..................................
Thus,

is true.

Therefore

is true for all

by the method of Mathemtaical Induction

If there is any step that confuses you, say so