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Math Help - Geometric progression

  1. #1
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    Geometric progression

    Hello guys, I need help solving the following question:

    The sum of the first three terms in a geometric progression is 195. The difference (?) between the first and second term is larger in 75 than the third term. Find the first term and the common ratio of the given progression.

    That is:
    I a1 + a2 + a3 = 195
    II a2 - a1 = a3 + 75
    a1=? q=?

    Could I please have the full solution? (because I reached a stage where I couldn't continue and didn't know what to do next)

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by loui1410 View Post
    Hello guys, I need help solving the following question:

    The sum of the first three terms in a geometric progression is 195. The difference (?) between the first and second term is larger in 75 than the third term. Find the first term and the common ratio of the given progression.

    That is:
    I a1 + a2 + a3 = 195
    II a2 - a1 = a3 + 75
    a1=? q=?

    Could I please have the full solution? (because I reached a stage where I couldn't continue and didn't know what to do next)

    Thanks in advance.
    Define
    a_n = a_1r^{n - 1}
    where r is the geometric ratio.

    So
    a_2 = a_1r
    and
    a_3 = a_1r^2

    So your conditions are
    a_1 + a_1 r + a_1 r^2 = 195
    and
    a_1 r - a_1 = a_1 r^2 + 75

    Thus we have two equations in two unknowns.

    The first condition reads:
    a_1(r^2 + r + 1) = 195
    and the second reads
    a_1(-r^2 + r - 1) = 75

    There are a variety of ways to attack this. This is one:
    Solve the top equation for a_1:
    a_1 = \frac{195}{r^2 + r + 1}

    and insert it into the bottom equation:
    \left ( \frac{195}{r^2 + r + 1} \right )(-r^2 + r - 1) = 75

    And finally, multiply both sides by r^2 + r + 1:
    195(-r^2 + r - 1) = 75(r^2 + r + 1)

    This is a quadratic in r, which I am sure you can solve.

    Note: For good or for ill I get that r is a complex number. I see no reason why this shouldn't be so, but I doubt you expected it. As I am not familiar with seeing problems like this turn out to be complex, I am wondering if there isn't some kind of typo?

    -Dan
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  3. #3
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    Thanks a lot for your help. No there's no typo, the r is supposed to have two solutions: 1/3 and -4/3. But the problem is that the discriminant of the quadratic equation is negative!

    -195r^2+195r-195=75r^2+75r+75
    270r^2-120r+270=0
    Δ =144-4*27*27=-2772


    Where's my mistake? I can't find it!
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  4. #4
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    Quote Originally Posted by loui1410 View Post
    Thanks a lot for your help. No there's no typo, the r is supposed to have two solutions: 1/3 and -4/3. But the problem is that the discriminant of the quadratic equation is negative!

    -195r^2+195r-195=75r^2+75r+75
    270r^2-120r+270=0
    Δ =144-4*27*27=-2772


    Where's my mistake? I can't find it!
    There's no mistake. Topsquark has already implied this technical infelicity:

    Quote Originally Posted by topsquark View Post
    [snip]
    Note: For good or for ill I get that r is a complex number. I see no reason why this shouldn't be so, but I doubt you expected it. As I am not familiar with seeing problems like this turn out to be complex, I am wondering if there isn't some kind of typo?

    -Dan
    which is why he asked if there's a typo.

    If there's no typo in what you posted, then the typo must be in the source of the question (textbook? problem sheet from class? Such things aren't infallible ......)

    Alternatively, Topsquark's (quite reasonable) mathematical interpretation of

    "The difference (?) between the first and second term is larger in 75 than the third term."

    could be different to what the source of the question intends .....
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  5. #5
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    It is supposed to be a_1-a_2=a_3+75, not a_2-a_1=a_3+75 as I thought. The language which was the question originally written in is not my mother-tongue, and English isn't my mother-tongue neither. So sorry guys, it was only a result of mistranslation Anyway, I reached the correct solution now.

    Thanks again.
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