Since k can be any positive integer, then what is the need for k+1?
Sk=1+2+3 ... +k+(k+1)
Sk+1=1+2+3 ... +k+(k+1)
They seem to be both saying the same thing?
Well, the way you have written them above, they are! And one statement does NOT follow from the other- indeed, they can't both be right
.
Did you mean write "Sk= 1+ 2+ 3+ ...+ k"?
In "proof by induction" you need to prove "If Sk is true (for
some k) then Sk+1 is true.
If you knew that "Sk is true for
all x" then it would be true for Sk+1 since that is simply a different value of k- but that is NOT what is being assumed. That is, by the way, why many people prefer to use a different symbol for the general statement than for the "induction step".
I presume you are using the example "Prove that 1+ 2+ 3+ ...+ n= n(n+1)/2"
To prove that "by induction on n", we first prove it is true when n= 1. That's typically easy- if we set n= 1, the left side is just 1 and the right side is 1(1+1)/2= 1(2)/2= 1. Yes, they are equal.
Now, we assume that, for some
single value of k, 1+ 2+ 3+ ...+ k= k(k+1)/2- that is the formula above with the general "n" replace by the specific "k". We are NOT assuming this for all k or all n- that would be what we are trying to prove. We are saying "IF" that is true for a specific value of k, then
1+ 2+ 3+ ...+ k+ (k+1)= k(k+1)/2+ (k+1). That is, I have taken the sum up to k and added (k+1) to it. We can factor out (k+1) and have (k+1)[k/2+ 1]= (k+1)(k/2+ 2/2)= (k+1)(k+2)/2= (k+1)((k+1)+1)/2. That is now the same formula with n replace by k+1.
The idea is that we have prove the statement for n= 1 separately. And we have proved "if it is true for n= k= 1, it is true for n= k+1= 2". Now that we know the statement is true for n= k= 2, it must be true for n= k+1= 3. And if it is true for n= k= 3, it is true for n= k+1= 4. .... The whole point of "proof by induction" is to avoid having to say all of that!