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Geometric Sequences

  1. May 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the first term in this geometric sequence that exceeds 500.
    2, 4, 8, 16, ...

    2. Relevant equations
    Un = arn-1


    3. The attempt at a solution
    a = 2, r = 2
    Un = 2 x 2n-1 > 500
    2 x (2n)(2-1) > 500
    log22 x log22n + log22-1 > log2500
    1 x n + (-1) > log2500
    n - 1 > log2500
    n > log2500 + log22
    n > log21000
    n > 9.96

    But this would make n = 10, whereas the answer should be n =9. What did I do wrong?!
     
  2. jcsd
  3. May 14, 2012 #2
    The log needs to be taken for the whole expression in the LHS. Why separate the LHS into multiple terms the first place?? :confused:
     
  4. May 14, 2012 #3
    The separation doesn't affect the calculation, it is the same.
    log22 x (n-1)log22 > log2500
    1 x (n-1) > log2500
    n - 1 > log2500
    n > log2500 + log22
    n > 9.96
     
  5. May 14, 2012 #4
    No no!

    I said you need to take the log for the whole LHS. Log is a function and just like, say sin, its taken for the whole expression, rather than each bit. Think of it this way,

    [itex]log(2).log(2^{n-1})[/itex]

    and

    [itex]log(2^{n})[/itex]

    Are they both the same?
     
    Last edited: May 14, 2012
  6. May 14, 2012 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    ln(2(2n-1))= ln(2)+ln(2n-1= 1+ n- 1= n.
    Or, more simply, 2(2n-1)= 21+n-1= 2n.

    That first "x" should be "+"

     
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