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anonymous

  • one year ago

Help with arithmetic?

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  1. anonymous
    • one year ago
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    @uri @SolomonZelman @mathstudent55

  2. anonymous
    • one year ago
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    What are the explicit equation and domain for an arithmetic sequence with a first term of 5 and a second term of 2?

  3. anonymous
    • one year ago
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  4. mathstudent55
    • one year ago
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    An arithmetic sequence has a common difference. Since the second term is 2 and the first term is 5, the common difference is 2 - 5 = -3 When you add the common difference to a term, you get the next term.

  5. mathstudent55
    • one year ago
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    5 5 + (- 3) = 2 2 + (- 3) = -1 -1 + (- 3) = -4 -4 + (-3) = -7 The first 5 terms of the sequence are: 5, 3, -1, -4, -7

  6. mathstudent55
    • one year ago
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    Since adding -3 is the same as subtracting 3, you can eliminate the first two choices because in those choices, you are subtracting multiples of 2. We need to subtract multiples of 3 to find subsequent terms. The answer has to be choice C or choice D.

  7. anonymous
    • one year ago
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    Right! Wow this is making so much more sense than I thought it would

  8. mathstudent55
    • one year ago
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    To figure out which one it is, look at choice C. \(a_n = 5 - 3(n - 1)\) for all integers n, such that \(n \ge 0\)

  9. mathstudent55
    • one year ago
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    Use the equation of choice C., and plug in the first value of n. That would be n = 0. \(a_n = 5 - 3(n - 1)\) \(a_0 = 5 - 3(0 - 1)\) What value do you get for \(a_0\) ?

  10. anonymous
    • one year ago
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    8!

  11. mathstudent55
    • one year ago
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    Correct. According to choice C., the first term in the sequence is called \(a_0\), and it is 8. We were told the first term is 5, so choice C cannot be correct.

  12. mathstudent55
    • one year ago
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    Wait. I got the two choices confused. I did choice D above, where n>= 0. Choice D. is eliminated.

  13. mathstudent55
    • one year ago
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    Now let's look at choice C., which is C. \(a_n = 5 - 3(n - 1)\) for all integers n, such that \(n \ge 1\) \(a_n = 5 - 3(n - 1)\) \(a_1 = 5 - 3(1 - 1)\) Now for choice C., what do you get for the first term?

  14. anonymous
    • one year ago
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    5

  15. mathstudent55
    • one year ago
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    Correct. Choice C. gives us the correct first term. Now notice what happens as n becomes 2, then 3, then 4, etc. Each time you are subtracting 1 from the next integer, then multiplying it by -3. That means first you subtract 0 from 5 (what you did for term 1) Then you subtract 3 from 5, then you subtract 6 from 5, then you subtract 9 from 5, etc. giving you the terms of the sequence. The answer is C.

  16. anonymous
    • one year ago
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    Okay, got it! Makes total sense. Thank you so SO much!

  17. anonymous
    • one year ago
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    Is my answer right with this one? It's a similar problem. @mathstudent55

  18. mathstudent55
    • one year ago
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    Let's see. Choice D. starts with n = 0. What do you get when you replace n with 0 below? \(a_n = 4(-12)^{n - 1} \) \(a_0 = 4(-12)^{0 - 1} \) What is \(a_0\) ?

  19. anonymous
    • one year ago
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    -.333?

  20. anonymous
    • one year ago
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    Oh sorry .333

  21. mathstudent55
    • one year ago
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    \(a_0 = 4(-12)^{0 - 1} = 4 (12)^{-1} = \dfrac{4}{12} = \dfrac{1}{3}\) Correct. We are told the first term is 4. That means this cannot be the answer.

  22. anonymous
    • one year ago
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    Yeah, I was hesitant on my answer. Glad I checked!

  23. anonymous
    • one year ago
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    So it would be C., correct?

  24. mathstudent55
    • one year ago
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    In the previous problem we had an arithmetic sequence. Notice this is a geometric sequence. Do you know the difference between a geometric sequence and an arithmetic sequence?

  25. anonymous
    • one year ago
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    I don not.

  26. mathstudent55
    • one year ago
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    In an arithmetic sequence (as we saw in the earlier problem) there is a common difference. If you add the common difference to a term, you get the next term. To find the common difference, subtract a term from the next term. In a geometric sequence there is a common ratio. If you multiply a term by the common ratio, you get the next term. To find the common ratio, divide a term by the previous term.

  27. mathstudent55
    • one year ago
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    Look at the first term and the second term: 4 and -8. What is -8/4 = ?

  28. anonymous
    • one year ago
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    Ooo, -2

  29. anonymous
    • one year ago
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    So would it be B?

  30. mathstudent55
    • one year ago
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    Right. The common ratio is -2, so you need an equation with a -2. That means we need A. or B.

  31. anonymous
    • one year ago
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    The sequence follows by -8, 16, -32, 64 . . .

  32. mathstudent55
    • one year ago
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    B. is correct because it gives us the correct first term, 4.

  33. anonymous
    • one year ago
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    Okay, just asking, but why wouldn't it be A.?

  34. mathstudent55
    • one year ago
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    Let's look at A. \(a_n =4(-2)^{n - 1} \); \(n \ge 0\) The first term uses n = 0: \(a_0 =4(-2)^{0 - 1} \) What is \(a_0\) using choice A.?

  35. anonymous
    • one year ago
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    -2

  36. mathstudent55
    • one year ago
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    Right, but we were told the first term is 4, so A. cannot be correct.

  37. anonymous
    • one year ago
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    Ooo okay. Got it! Thank you so much!

  38. anonymous
    • one year ago
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    I got 100%!! Thanks so much!

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