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    I always get stuck on the k+1 part

    I have no idea what to do once I reach that stage, and this has cost me many marks in practise papers

    Please help! =[
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    We can't help unless you give us the question and show your working.
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    The idea is that you're told to prove that something is true for all n. You check the base case (usually n=0 or 1), and then you assume it's true for n=k. What you then have to do is, assuming no more than that it is true for n=k, follow a sequence of logical steps to show that, if it's true for n=k, then it's true for n=k+1.

    For example say we have to show that \sin n\pi = 0 for all n \ge 0. Clearly \sin (0\pi) = \sin 0 = 0 so the base case is fine. Next we assume that \sin k\pi = 0. Now, we can write \sin (k+1)\pi = \sin (k\pi + \pi) = \sin k\pi \cos \pi + \cos k \pi \sin \pi. The first term is zero by our assumption that \sin k \pi = 0, and the second term is zero because we know that \sin \pi = 0, and so assuming truth for n=k implies truth for n=k+1. That's the inductive step, so we're done.
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    A general hint for induction problems: You usually want to show  f(n)=g(n) . What you need to do is find a link between  f(n+1) and  f(n) . If it's a summation problem you will probably want to consider  f(n+1)-f(n) , if it's a product problem it might be  f(n+1)/f(n) and so on.
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    (Original post by nuodai)
    The idea is that you're told to prove that something is true for all n. You check the base case (usually n=0 or 1), and then you assume it's true for n=k. What you then have to do is, assuming no more than that it is true for n=k, follow a sequence of logical steps to show that, if it's true for n=k, then it's true for n=k+1.

    For example say we have to show that \sin n\pi = 0 for all n \ge 0. Clearly \sin (0\pi) = \sin 0 = 0 so the base case is fine. Next we assume that \sin k\pi = 0. Now, we can write \sin (k+1)\pi = \sin (k\pi + \pi) = \sin k\pi \cos \pi + \cos k \pi \sin \pi. The first term is zero by our assumption that \sin k \pi = 0, and the second term is zero because we know that \sin \pi = 0, and so assuming truth for n=k implies truth for n=k+1. That's the inductive step, so we're done.
    this is very confusing to me :/, just the k+1 part.
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    http://i.imgur.com/h2ahK.png

    for example here, i know on the right side he just replaced k with k+1, but on the left side why does he turn (3k-4) into (3k-4) + (3(k+1)-4)

    are you meant to leave the (3k-4) in the lhs THEN add another value that is with the k's replaced with k+1?

    thanks
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    (Original post by TimetoSucceed)
    http://i.imgur.com/h2ahK.png

    for example here, i know on the right side he just replaced k with k+1, but on the left side why does he turn (3k-4) into (3k-4) + (3(k+1)-4)

    are you meant to leave the (3k-4) in the lhs THEN add another value that is with the k's replaced with k+1?

    thanks
    You're trying to show that, for all n, -1 + 2 + 5 + 8 + \cdots + (3n-4) = \frac{n}{2}(3n-5). Now, the LHS in the case n=k looks like:

    -1 + 2 + 5 + 8 + \cdots + (3k-4)

    The goal is to show that the n=k+1 sum is equal to what you get when you replace 'n' by 'k+1' in the formula for what you're trying to show. The LHS in the case n=k+1 looks like:

    -1 + 2 + 5 + 8 + \cdots + (3(k+1)-4)

    But inside the \cdots is hidden a load of other terms, and 3k-4 is one of them (it's just the kth term in the sum), because to get to the n=k+1 sum you add the (k+1)th term to the n=k sum, right? And so to emphasise what you're meant to do, it's been written as

    -1 + 2 + 5 + 8 + \cdots + (3k-4) + (3(k+1)-4)

    What he's put on the RHS is what you need to try to show is true, using only your assumption about the n=k case. Your assumption about the n=k case tells you that

    -1+2+5+8+\cdots+(3k-4) = \frac{k}{2}(3k-5)

    and so you can write your n=k+1 sum as:

    \frac{k}{2}(3k-5) + (3(k+1)-4)

    (Because to get from the n=k sum to the n=k+1 sum you just add the (k+1)th term to both sides)

    You then have to rearrange this to show that it's equal to \frac{k+1}{2}(3(k+1)-5).

    I've tried to make this as clear and obvious as I can, but if you still don't understand, let me know (but refer to specific parts rather than 'the k+1 bit').
 
 
 
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