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Math Help - Geometric progression involving logs!

  1. #1
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    Geometric progression involving logs!

    Hey, can anyone help me with this please?

    5. In a geometric progression, the first term is 5 and the second term is 4.8
    i) I have solved this. r = 0.96
    ii) The sum of the first n terms is greater than 124. Show that

    0.96^n < 0.008,

    and use logarithms to calculate the smallest possible value of n.

    Thank you if you can help me with this.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by LoveDeathCab View Post
    Hey, can anyone help me with this please?

    5. In a geometric progression, the first term is 5 and the second term is 4.8
    i) I have solved this. r = 0.96
    a geometric progression is one in which the terms are given by the formula:
    a_n = ar^(n - 1) for n = 1,2,3,4,5...
    where a_n is the nth term, a is the first term and r is the common ration

    we are told that the first term of a particular geometric sequence is 5 and the second term is 4.8

    now r is given by r = (a_{n+1})/(a_n)
    in particular, r = (a_2)/(a_1) = 4.8/5 = 0.96

    you are correct

    ii) The sum of the first n terms is greater than 124. Show that

    0.96^n < 0.008,

    and use logarithms to calculate the smallest possible value of n.
    The sum of the first n terms of a geometric sequence/progression is given by:
    S_n = a(1 - r^n)/(1 - r)
    we are told that S_n > 124
    => a(1 - r^n)/(1 - r) > 124
    => 5(1 - (0.96)^n)/(1 - 0.96) > 124
    =>5(1 - (0.96)^n)/(0.04) > 124
    => 5(1 - (0.96)^n) > 4.96 ..............i multiplied through by 0.04
    => 1 - (0.96)^n > 0.992 ..................i divided through by 5
    => -(0.96)^n > -0.008 ....................i subtracted 1 from both sides
    => (0.96)^n < 0.008 ...............................i multiplied through by -1, so i flipped the inequality sign.

    Now we will use logs to find the smallest possible n.

    0.96^n < 0.008
    take log to the base 10 of both sides
    => log(0.96^n) < log(0.008)
    => nlog(0.96) < log(0.008)
    => n > log(0.008)/log(0.96) ...........note that log(0.96) is negative, so when i divided by it, i flipped the inequality sign
    => n > 118.27

    so the smallest possible value for n is 119
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