The steps for proof for mathematical induction induction for summation of a series are listed in my book as:
1. prove that it is true for n=1.
2. assume that it is true for n=k.
3. thus prove that it is true for n=k+1

Now it is step 2 and 3 that bothers me. We assume that the expression is true for n=k. So the expression can also be equal to r. And r can be equal to k+1. So is we assume step 2, we assume step 3 as well, there is must not be a need to prove it.

In other words, for step 2, k can equal any positive integer, so we are assuming the thing that we should have proved.

I hope I have clearly explained why I am confused. Obviously there is a logical explanation for this, which I hope will be given by a reader.

Thank you for taking time to read this.

The thing is that it may not be true for any random value of k. For example, suppose that we are asked to prove that the sum of 1 to n is n - you probably already know that this sum actually equals n(n+1)/2, so the hypothesis is wrong. But it is certainly true for n = 1. Next we assume it's true for some value of n=k, and seek to prove that it must also be true for n=k+1:

\sum _{i=1}^{k+1} i = \sum _{i=1}^k i+ (k+1) = k + (k+1 ) = 2k+1

Since 2k+1 does not equal n=k+1, we have disproved the hypothesis. So you see we didn't jump from an assumption that if true for n= k it must also be true for n=k+1.

It always takes a few word's from you to clear all my confusions! Thanks. :-)