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Binomial Theorem

  1. Aug 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Evaluate
    [tex]\sum^{m}_{r=0} ^{ n + r }C_{n}[/tex]

    I can handle things when the lower thing in the combination part is changing, what shall I do with this one?
     
  2. jcsd
  3. Aug 16, 2008 #2

    rock.freak667

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    Homework Helper

    Try writing out a few terms in the series and see if it helps.
     
  4. Aug 16, 2008 #3
    I get this:
    [tex]^{n}C_{n} + ^{n+1}C_{n} + ^{n+2}C_{n} + ........ + ^{n+r}C_{n}
    [/tex]
    [tex]^{n}C_{0} + ^{n+1}C_{1} + ^{n+2}C_{2} + ........ + ^{n+r}C_{r}
    [/tex]

    All I can do is this, now both the superscript and th subscript are increasing in A.P.
     
  5. Aug 17, 2008 #4
    The thread is still unsolved....
     
  6. Aug 17, 2008 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    the suggestion was that you actually look at a few specific examples.
    If m= 1, you have
    [tex]^nC_n+ ^{n+1}C_n= \frac{n!}{n!0!}+ \frac{(n+1)!}{n!1!}= 1+ n+ 1= n+ 2[/tex]
    If m= 2, you have
    [tex]^nC_n+ ^{n+1}C_n+ ^{n+2}C_n= n+ 2+ \frac{(n+2)!}{n! 2!}= n+ 2+ (n+1)(n+2)/2= n+2+ \frac{1}{2}n^2+ \frac{3}{2}x+ 1= \frac{1}{2}n^2+ \frac{5}{2}n+ 3[/tex]

    Try a few more like that and see if anything comes to mind.
     
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