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Math Help - Very basic induction (sum of first n positive integers) - help needed

  1. #1
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    Very basic induction (sum of first n positive integers) - help needed

    Whenever someone has a moment, can you answer this? It is the very first example in the book: http://i.imgur.com/lxbdp.jpg
    (I underlined the part in red in the picture)
    This is the only part I couldn't understand right from the beginning.

    In the inductive step, when they are writing the formula for P(k+1), how did they get it?

    On the right side all clear - they replaced k with k+1 as I would expect.

    But on the left side there's still k and they added (k+1)? I don't get that.
    Why didn't they replace k with k+1, like they did on the right side?

    Thanks.
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  2. #2
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    Re: Very basic induction (sum of first n positive integers) - help needed

    Well the sum of the first k+1 positive integers is the sum of the first k positive integers plus the number (k+1).

    So P(k+1)=P(k)+(k+1)=\frac{k(k+1)}{2}+(k+1)~.
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  3. #3
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    Re: Very basic induction (sum of first n positive integers) - help needed

    Strictly speaking, P(k) is a proposition (equality in this case), not a number. The left-hand side of P(k) is \sum_{i=1}^k i, or, in words, the sum of all numbers from 1 to k. In writing P(k+1) they did replace k with k+1 in both sides of the equality P(k). The left-hand side became \sum_{i=1}^{k+1} i, or 1+\dots+k+(k+1). In words, "the sum of all numbers from 1 to k+1" mean the same thing.
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  4. #4
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    Re: Very basic induction (sum of first n positive integers) - help needed

    Thanks, that does make it clear. Also explains several other problems I was having trouble with.
    I was thinking in terms of "replace k by k+1 in the expression", but now I see that I need to treat P(k+1) as a proposition, i.e. go back to the part where the proposition was defined (in text), and put k+1 there.
    Thanks again, I very much appreciate your help. That was my first post, and I'm feeling like this place is going to be a great asset.
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