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kathert Group Title

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

  • one year ago
  • one year ago

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

    • one year ago
  2. theEric Group Title
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    http://www.mathsisfun.com/algebra/sequences-sums-geometric.html

    • one year ago
  3. kathert Group Title
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    But what do I plug in there to get the answers?

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

    • one year ago
  5. cwrw238 Group Title
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    r = common ratio = second term / first term a1 = first term and n = number of terms

    • one year ago
  6. cwrw238 Group Title
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    so for question 1 r = -16/8 = -2 a1 = 8 and n = 15

    • one year ago
  7. kathert Group Title
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    So would the answer for the first one be 87384?

    • one year ago
  8. theEric Group Title
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    @kathert I agree with 87384 :)

    • one year ago
  9. cwrw238 Group Title
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    8 * ( 1 - (-2)^15) ------------- = 87384 1 - (-2)

    • one year ago
  10. kathert Group Title
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    Oh yay! I got it right! Sorry I suck at math...

    • one year ago
  11. cwrw238 Group Title
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    A GS can be written as a, ar , ar^2 , ar^3 .......

    • one year ago
  12. theEric Group Title
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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 .

    • one year ago
  13. kathert Group Title
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    @theEric Thanks :)

    • one year ago
  14. cwrw238 Group Title
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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

    • one year ago
  15. kathert Group Title
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    Whoa how do I do 7 root in my calculator??

    • one year ago
  16. theEric Group Title
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    Do it like this: r ^ (- 7)

    • one year ago
  17. theEric Group Title
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    No!

    • one year ago
  18. theEric Group Title
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    Bad me!

    • one year ago
  19. theEric Group Title
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    r ^ (1/7)

    • one year ago
  20. theEric Group Title
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    That's what you want, \[\Huge r^{\frac{1}{7}}\]

    • one year ago
  21. kathert Group Title
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    Haha okay gimme a minute to figure this out...

    • one year ago
  22. kathert Group Title
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    I got 39364 for number 2 is that right?

    • one year ago
  23. kathert Group Title
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    Okay okay how do i do the last two??

    • one year ago
  24. theEric Group Title
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    That's what I got for #2 as well.

    • one year ago
  25. theEric Group Title
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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 \(\Large a\frac{1-r^n}{1-r}\).

    • one year ago
  26. theEric Group Title
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    So, you have \(a\) and \(r\), and you need \(n\) or \(r^n\).

    • one year ago
  27. theEric Group Title
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    Do you see how to get that?

    • one year ago
  28. kathert Group Title
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    uhhh.... no not really...so it would be -11 (1-(-4^n)/(1--4)?

    • one year ago
  29. theEric Group Title
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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.

    • one year ago
  30. theEric Group Title
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    General guideline: if you want something, solve for it.

    • one year ago
  31. kathert Group Title
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    wait so n is -11,246??

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

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

    • one year ago
  34. theEric Group Title
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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\[\frac{11\ r^n}{11}=\frac{-11,264}{11}\]\[\frac{\cancel{11}\ r^n}{\cancel{11}}=\frac{-11,264}{11}\]\[r^n=\frac{-11,264}{11}\]

    • one year ago
  35. theEric Group Title
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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.

    • one year ago
  36. kathert Group Title
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    Ohhhhh okay! I get it now!

    • one year ago
  37. theEric Group Title
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    SWEET! :) So, we'll both calculate #3 and see what we get....

    • one year ago
  38. kathert Group Title
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    I got -11 1/5

    • one year ago
  39. theEric Group Title
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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.

    • one year ago
  40. kathert Group Title
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    okay

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

    • one year ago
  42. theEric Group Title
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    Maybe you just had some calculator error.

    • one year ago
  43. kathert Group Title
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    Oh I see what I did wrong

    • one year ago
  44. kathert Group Title
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    How would I go about starting the last one?

    • one year ago
  45. theEric Group Title
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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?

    • one year ago
  46. theEric Group Title
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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.

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

    • one year ago
  48. kathert Group Title
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    so would it be 10 (1-(781250/10))/1-r?

    • one year ago
  49. kathert Group Title
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    @theEric

    • one year ago
  50. theEric Group Title
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    Sorry! Hi!

    • one year ago
  51. kathert Group Title
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    Its alright I disappeared for dinner so... haha

    • one year ago
  52. theEric Group Title
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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\[\sqrt[7]{r^7}=r=\sqrt[7]{78,125}=5\]

    • one year ago
  53. theEric Group Title
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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 \(r\).

    • one year ago
  54. theEric Group Title
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    Which you can rearrange to spell \(r\ a\ n\): fun fact..

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

    • one year ago
  56. theEric Group Title
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    Haha, thanks! And check your "\(r^n\)", or \(5^8\).

    • one year ago
  57. kathert Group Title
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    so 390625?

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

    • one year ago
  59. kathert Group Title
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    so that goes where i put the 78125

    • one year ago
  60. theEric Group Title
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    Yep!

    • one year ago
  61. theEric Group Title
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    It is \(r^n\), after all.

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

    • one year ago
  63. kathert Group Title
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    I got 781,248 for my answer

    • one year ago
  64. theEric Group Title
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    I got the same! :) Congrats!

    • one year ago
  65. theEric Group Title
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    Any questions about this problem?

    • one year ago
  66. kathert Group Title
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    Nope I'm good! Thank you for your help!

    • one year ago
  67. theEric Group Title
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    You're welcome! Take care!

    • one year ago
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