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kathert

Please help... Only 2 days left for me on a summer school class and I am lost... 1. What is the sum of the geometric sequence 8, -16, 32 . if there are 15 terms? (1 point) 2. What is the sum of the geometric sequence 4, 12, 36 . if there are 9 terms? (1 point) 3. What is the sum of a 6-term geometric sequence if the first term is 11, the last term is -11,264 and the common ratio is -4? (1 point) 4. What is the sum of an 8-term geometric sequence if the first term is 10 and the last term is 781,250? (1 point) Show all work as well

  • 9 months ago
  • 9 months ago

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  1. cwrw238
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    the formula for sum of n terms is Sn = a1 * (1 - r^n) ------ 1 - r

    • 9 months ago
  2. theEric
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    http://www.mathsisfun.com/algebra/sequences-sums-geometric.html

    • 9 months ago
  3. kathert
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    But what do I plug in there to get the answers?

    • 9 months ago
  4. theEric
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    The link I posted will walk you through what geometric sequences are, and that formula to get the answers!

    • 9 months ago
  5. cwrw238
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    r = common ratio = second term / first term a1 = first term and n = number of terms

    • 9 months ago
  6. cwrw238
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    so for question 1 r = -16/8 = -2 a1 = 8 and n = 15

    • 9 months ago
  7. kathert
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    So would the answer for the first one be 87384?

    • 9 months ago
  8. theEric
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    @kathert I agree with 87384 :)

    • 9 months ago
  9. cwrw238
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    8 * ( 1 - (-2)^15) ------------- = 87384 1 - (-2)

    • 9 months ago
  10. kathert
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    Oh yay! I got it right! Sorry I suck at math...

    • 9 months ago
  11. cwrw238
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    A GS can be written as a, ar , ar^2 , ar^3 .......

    • 9 months ago
  12. theEric
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    I've \(always\) been slow at math. But it took practice, and getting help, and I can do more math things now! Best of luck in your class! :) And it looks like you're in good hands with cwrw238 .

    • 9 months ago
  13. kathert
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    @theEric Thanks :)

    • 9 months ago
  14. cwrw238
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    for the last probem you can find the common ratio r by dividing the 8th term by the first then you take the 7th root 8th term = ar^7 8th term / first term = ar^7 / a = r^7 781,250 / 10 = 78125 now use your calculator to find the 7th root of 78125 then use the sum formula gotta go now

    • 9 months ago
  15. kathert
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    Whoa how do I do 7 root in my calculator??

    • 9 months ago
  16. theEric
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    Do it like this: r ^ (- 7)

    • 9 months ago
  17. theEric
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    No!

    • 9 months ago
  18. theEric
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    Bad me!

    • 9 months ago
  19. theEric
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    r ^ (1/7)

    • 9 months ago
  20. theEric
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    That's what you want, \[\Huge r^{\frac{1}{7}}\]

    • 9 months ago
  21. kathert
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    Haha okay gimme a minute to figure this out...

    • 9 months ago
  22. kathert
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    I got 39364 for number 2 is that right?

    • 9 months ago
  23. kathert
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    Okay okay how do i do the last two??

    • 9 months ago
  24. theEric
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    That's what I got for #2 as well.

    • 9 months ago
  25. theEric
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    3. What is the sum of a 6-term geometric sequence if the first term is 11, the last term is -11,264 and the common ratio is -4? (1 point) Well, the first term is your \(a\).\[a=11\]The last term is your \(a\ r^n\).\[a\ r^n=-11,264\]The common ratio is your \(r\).\[r=-4\] You need \(\Large a\frac{1-r^n}{1-r}\).

    • 9 months ago
  26. theEric
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    So, you have \(a\) and \(r\), and you need \(n\) or \(r^n\).

    • 9 months ago
  27. theEric
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    Do you see how to get that?

    • 9 months ago
  28. kathert
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    uhhh.... no not really...so it would be -11 (1-(-4^n)/(1--4)?

    • 9 months ago
  29. theEric
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    Yep! Hey, you know \(a\ r^n=-11,264\), so you can solve for \(r^n\)! That's how you'll finish that problem.

    • 9 months ago
  30. theEric
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    General guideline: if you want something, solve for it.

    • 9 months ago
  31. kathert
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    wait so n is -11,246??

    • 9 months ago
  32. theEric
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    No, \(a\ r^n=-11,264\). So you divide both sides by \(a=11\). That's algebra!

    • 9 months ago
  33. kathert
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    im sorry but im so lost.... where does the -11246 come in? do i set it equal to the equation?

    • 9 months ago
  34. theEric
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    It is necessary to find the \(r^n\). Let me show you. Are you familiar with algebra? \[a\ r^n = -11,264\]and\[a=11\]By substituting \(11\) for \(a\), which is okay because it's the same value either way, you'll get:\[11\ r^n=-11,264\]Now, you want to get \(r^n\) alone. So what you do is, you divide both sides by \(11\). 1. If the two sides are equal, and you do the same thing to both sides, both sides will still be equal! 2. Why divide by \(11\)? Well \(r^n\) is being multiplied by \(11\), and so you want to negate that. You want to make it be \(\Large \frac{\cancel{11}\ r^n}{\cancel{11}}\) So, we left our equation off at\[11\ r^n=-11,264\]We divide by \(11\) to get\[\frac{11\ r^n}{11}=\frac{-11,264}{11}\]\[\frac{\cancel{11}\ r^n}{\cancel{11}}=\frac{-11,264}{11}\]\[r^n=\frac{-11,264}{11}\]

    • 9 months ago
  35. theEric
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    Since you now have \(r^n\), \(a\), and \(r\), you can use that formula that you used for #1 and #2.

    • 9 months ago
  36. kathert
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    Ohhhhh okay! I get it now!

    • 9 months ago
  37. theEric
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    SWEET! :) So, we'll both calculate #3 and see what we get....

    • 9 months ago
  38. kathert
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    I got -11 1/5

    • 9 months ago
  39. theEric
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    I got \(2255\)... Let me use Wolfram Alpha to double check. Then I can show you a link to the math.

    • 9 months ago
  40. kathert
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    okay

    • 9 months ago
  41. theEric
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    http://www.wolframalpha.com/input/?i=11*%281-%28-11264%2F11%29%29%2F%281-%28-4%29%29

    • 9 months ago
  42. theEric
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    Maybe you just had some calculator error.

    • 9 months ago
  43. kathert
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    Oh I see what I did wrong

    • 9 months ago
  44. kathert
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    How would I go about starting the last one?

    • 9 months ago
  45. theEric
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    Well, I'm sure you know the formula you need to use, by now!\[\text{sum}=a\frac{(1-r^n)}{(1-r)}\] 4. What is the sum of an 8-term geometric sequence if the first term is 10 and the last term is 781,250? (1 point) You need \(a\), \(r\), and \(r^n\), or \(n\). What do you know from the problem, about the geometric sequence?

    • 9 months ago
  46. theEric
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    Refresher: \(a\) is the first term, or the common multiplier. \(r\) is the common ratio. \(n\) is the number of terms in the sequence.

    • 9 months ago
  47. theEric
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    8-term \(\rightarrow n=8\) first term is 10 \(\rightarrow a=10\) last term is 781,250 \(\rightarrow a\ r^{n-1} =781,250\)

    • 9 months ago
  48. kathert
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    so would it be 10 (1-(781250/10))/1-r?

    • 9 months ago
  49. kathert
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    @theEric

    • 9 months ago
  50. theEric
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    Sorry! Hi!

    • 9 months ago
  51. kathert
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    Its alright I disappeared for dinner so... haha

    • 9 months ago
  52. theEric
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    Nope, sorry! Small mistake! \[a\ r^{n-1} =781,250\]\(\qquad\Downarrow\qquad\)Substitute \(8\) for \(n\) \[a\ r^{8-1} =a\ r^{7}=781,250\]\(\qquad\Downarrow\qquad\)Divide both sides by \(a\), and then substitute \(10\) in for \(a\) \[r^7=\frac{781,250}{a}=\frac{781,250}{10}=78,125\]\(\qquad\Downarrow\qquad\)Get the seventh root of both sides\[\sqrt[7]{r^7}=r=\sqrt[7]{78,125}=5\]

    • 9 months ago
  53. theEric
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    Well, you knew you didn't know \(r\), so I guess your only mistake was substituting \(r^n\) with \(r^{n-1}=781,250\), but you definitely had the right idea otherwise! I just found \(r\) for you, then... Any questions on that part? Now you have \(a\), \(n\), and \(r\).

    • 9 months ago
  54. theEric
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    Which you can rearrange to spell \(r\ a\ n\): fun fact..

    • 9 months ago
  55. kathert
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    so it would be 8 *(1-78125)/(1-5) just to be sure nice fun fact btw haha

    • 9 months ago
  56. theEric
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    Haha, thanks! And check your "\(r^n\)", or \(5^8\).

    • 9 months ago
  57. kathert
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    so 390625?

    • 9 months ago
  58. theEric
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    You got \(78125\) from \(r^{n-1}\), so I see where that came from :) And, yep! \(390625\).

    • 9 months ago
  59. kathert
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    so that goes where i put the 78125

    • 9 months ago
  60. theEric
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    Yep!

    • 9 months ago
  61. theEric
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    It is \(r^n\), after all.

    • 9 months ago
  62. theEric
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    and your formula is\[\text{sum}=a\frac{(1-r^n)}{(1-r)}\]

    • 9 months ago
  63. kathert
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    I got 781,248 for my answer

    • 9 months ago
  64. theEric
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    I got the same! :) Congrats!

    • 9 months ago
  65. theEric
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    Any questions about this problem?

    • 9 months ago
  66. kathert
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    Nope I'm good! Thank you for your help!

    • 9 months ago
  67. theEric
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    You're welcome! Take care!

    • 9 months ago
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