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Sum of Getometric Sequence with alternating signs

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data

    5^2 - 5^3 + 5^4 - ... + (-1)^k*5^k whre k is an integer with k >= 2

    2. Relevant equations

    3. The attempt at a solution

    I know (5^(k-1) - 5^2)/2 gives you the sum if they were all positive. I tried multiplying it by (-1)^k or something but that just changes the sign. I wish I could give you more but I can't.
  2. jcsd
  3. Oct 18, 2011 #2


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    A geometric sequence is [itex]\sum_{n=0}^N ar^n[/itex]. And the sum is:
    [tex]\frac{1- r^{N+1}}{1- r}[/tex]
    r does not have to be positive. Your sequence has a= 1, r= -5.
  4. Oct 18, 2011 #3
    [tex]\frac{1- (-5)^{2+1}}{1- (-5)}[/tex] = 21 though and not 25
    Last edited: Oct 18, 2011
  5. Oct 18, 2011 #4


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    The sum that HallsofIvy gave includes (-5)0 and (-5)1
  6. Oct 18, 2011 #5


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    There are two ways to handle the fact that your sum starts with [itex]r^2[/itex] rather than [itex]r^0= 1[/itex].

    1) Factor out an [itex]r^2[/itex] [itex](-5)^2+ (-5)^3\cdot\cdot\cdot+ (-5)^k= (-5)^2(1+ (-5)+ \cdot\cdot\cdot+ (-5)^{k-2})[/itex]

    Use the formula I gave with n= k- 2 and then multiply by [itex](-5)^2= 25[/itex].

    2) Use the formula with n= k and then subtract of [itex](-5)^0+ (-5)^1= 1- 6= -4[/itex].
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