Results 1 to 2 of 2

Thread: Geometric Progression

  1. #1
    Feb 2007

    Geometric Progression

    Q1 a G.P is such that S2 = 60 and S3 = 114 find the first 3 terms

    Q2 8000 tons of a particular mineral were mined in each of the first 3 yrs of a mines operation. in the fourth yr the quantity mined was 90% of the third year's output, the fifth yr was 90% of the fourth yr and so on. the mine was closed at the end of the first year in which the amount mined fell below 1900 tons
    a) for how many yrs did the mine stay open?
    b) what total tonnage of the mineral was obtained from the mine during its lifetime?

    *i dont even understand wot the Question is asking
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    May 2006
    Lexington, MA (USA)
    Hello, Fibonacci!

    1) A G.P is such that $\displaystyle S_2 = 60\text{ and }S_3 = 114$. . Find the first 3 terms
    Sum of the first $\displaystyle n$ terms: .$\displaystyle S_n \;=\;a\,\frac{r^n-1}{r-1}$

    $\displaystyle S_3=114\quad\Rightarrow\quad a\,\frac{r^3-1}{r-1} \:=\:114\;\;{\color{blue}[1]}$

    $\displaystyle S_2 = 60 \quad\Rightarrow\quad a\,\frac{r^2-1}{r-1} \:=\:60\;\;{\color{blue}[2]}$

    Divide [1] by [2]: .$\displaystyle \frac{a\,\frac{r^3-1}{r-1}}{a\,\frac{r^2-1}{r-1}} \;=\;\frac{114}{60} \quad\Rightarrow\quad \frac{r^3-1}{r^2-1} \:=\:\frac{19}{10}$

    . . $\displaystyle \frac{(r-1)(r^2+r+1)}{(r-1)(r+1)} \:=\:\frac{19}{10} \quad\Rightarrow\quad \frac{r^2+r+1}{r+1} \:=\:\frac{19}{10}$

    . . which simplifies to: .$\displaystyle 10r^2 - 9r - 9 \:=\:0 \quad\Rightarrow\quad (3r-3)(5r+3) \:=\:0$

    . . Hence: .$\displaystyle r \;=\;\frac{3}{2},\;-\frac{3}{5}$

    But since two consecutive terms of the G.P. are positive, $\displaystyle r$ cannot be negative.

    . . Therefore: .$\displaystyle \boxed{r \;=\;\frac{3}{2}}$

    Substitute into [2]: .$\displaystyle a\,\frac{\left(\frac{3}{2}\right)^2 - 1}{\frac{3}{2}-1} \:=\:60 \quad\Rightarrow\quad\frac{5}{2}a \:=\:60 \quad\Rightarrow\quad\boxed{ a \:=\:24}$

    Therefore, the first three terms are: .$\displaystyle 24,\:36,\:54$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Geometric progression
    Posted in the Math Topics Forum
    Replies: 3
    Last Post: Apr 11th 2011, 05:06 AM
  2. Geometric Progression
    Posted in the Algebra Forum
    Replies: 4
    Last Post: Mar 17th 2010, 04:08 AM
  3. Geometric progression
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Jan 12th 2010, 04:22 AM
  4. Geometric Progression or Geometric Series
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: Oct 8th 2009, 07:31 AM
  5. Replies: 8
    Last Post: Mar 23rd 2009, 07:26 AM

Search Tags

/mathhelpforum @mathhelpforum