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ALmost got it, mathematical induction, writing terms seperatley

  1. Sep 30, 2006 #1
    Hello everyone i'm having problems on this last part of mathematical induction. I have to show that the two equations are equal to eachother. The book shows a few examples which i will show below. They are writing the kst term separately from the first k terms.

    Heres my problem firstly:
    Prove by mathematical induction:

    My Goal is to prove that those 2 equations do infact equal eachother, the boxed equations. Once I write the k term separatley then I go on to substitute from the inductive hypotheiss, then do some algebra.

    Examples of writing the terms seperately:

    Under my problem i attempted to mimic what they are doing, is that it or no? Any help would be great on explaining what they are doing here!

    Last edited: Sep 30, 2006
  2. jcsd
  3. Sep 30, 2006 #2
    it should be

    [tex]\sum_{i = 1}^{k + 2} i2^i = \left(\sum_{i = 1}^{k+1} i2^i \right) + (k+2)2^{k+2} [/tex]

    As a simple example consider:

    [tex] \sum_{i=1}^{4}i = 1 + 2 + 3 + 4 = (1 + 2 + 3) + 4 = \left(\sum_{i=1}^{3}i \right) + 4 [/tex]

    more generally, if f is a function defined on the integers, and a, b, and c are integers with [itex]a \leq b \leq c [/itex] then,

    \sum_{i = a}^{c} f(i) &= f(a) + f(a+1) + \ldots + f(b-1) + f(b) + f(b+1) + \ldots + f(c-1) + f(c) \\
    &= \left( f(a) + f(a+1) + \ldots + f(b-1) \right) + \left( f(b) + f(b+1) + \ldots + f(c-1) + f(c) \right) \\
    &= \left( \sum_{i = a}^{b-1} f(i) \right) + \left( \sum_{i = b}^{c} f(i) \right)

    in regards to your problem, f is the function
    [tex]f(k) = k2^k[/tex]
    and a = 1, b = c = k + 2

    Your proof up to this point looks great by the way, very nicely written.
    Last edited: Sep 30, 2006
  4. Oct 2, 2006 #3
    Thanks nocturnal, that explanation was very helpful. Sorry about the delayed responce.

    I'm now having the following troubles, i can't seem to manipulate the left hand side of the equation to look like the right hand side.

    This is my work:


    I'm suppose to get that to look like right hand side of the equation, (k+1)*2^(k+3) + 2, any help or suggestions would be great! I tried to factor and expand, and they dont' seem to work or maybe i can't see somthing.

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