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Math Help - Give a recursive definition

  1. #1
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    Give a recursive definition

    Hello! i am stuck on this recursive definition. I don't know what to do next.

    Q: Give a recursive definition of the sequence {an}, n = 1, 2, 3, … if
    a.) an = 4n – 2
    b.) an = 1 + (-1)^(n)
    c.) an = n(n + 1)
    d.) an = n
    __________________________________________________ _____

    I have this:

    a.)

    an = 4n – 2
    a1 = 4(1) – 2
    a1 = 4 – 2
    a1 = 2

    an = 4n – 2
    a2 = 4(2) – 2
    a2 = 8 - 2
    a2 = 6

    an = 4n – 2
    a3 = 4(3) – 2
    a3 = 12 - 2
    a3 = 10

    an = 4n – 2
    a4 = 4(4) – 2
    a4 = 16 - 2
    a4 = 14

    I can see they go up by 4 for part a, but do i leave it like this?

    b.)

    an = 1 + (-1)^(n)
    a1 = 1 + (-1)^(1)
    a1 = 1 – 1
    a1 = 0

    an = 1 + (-1)^(n)
    a2 = 1 + (-1)^(2)
    a2 = 1 + 1
    a2 = 2

    an = 1 + (-1)^(n)
    a3 = 1 + (-1)^(3)
    a3 = 1 – 1
    a3 = 0

    an = 1 + (-1)^(n)
    a4 = 1 + (-1)^(4)
    a4 = 1 + 1
    a4 = 2

    I can see the sequence going 0, 2, 0, 2, ... etc.

    c.)

    an = n(n + 1)
    a1 = 1(1 + 1)
    a1 = 1(2)
    a1 = 2

    an = n(n + 1)
    a2 = 2(2 + 1)
    a2 = 2(3)
    a2 = 6

    an = n(n + 1)
    a3 = 3(3 + 1)
    a3 = 3(4)
    a3 = 12

    an = n(n + 1)
    a4 = 4(4 + 1)
    a4 = 4(5)
    a4 = 20

    I am not sure on this one?
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  2. #2
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    For part a, the sequence is 2,\;6,\;10,\;....
    a_1 =2 and n \geqslant 2\;,\,a_n  = a_{n - 1}  + 4.

    Now you show us part b.
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  3. #3
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    Plato, I don't understand this:

    and .

    I got the sequence of 2, 6, 10, etc. But i don't understand the above part of what you wrote?
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  4. #4
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    Quote Originally Posted by Grillakis View Post
    Plato, I don't understand this:

    and .

    I got the sequence of 2, 6, 10, etc. But i don't understand the above part of what you wrote?
    Are you saying that you do not understand what a recursively defined sequence is?
    If so, why are you asked to do this problem?
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  5. #5
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    Well reason why im doing this is because its part of my HW and I missed his class when he supposedly covered this. I was sick so i am left to understand this on my own. He posts his lecture notes online and that is what I am going over. When I am looking through the textbook I am trying to understand it and when i can't I usually search online and I try to understand that this way or through forums, hopefully trying to get some help on how to do this. And if this isn't bad enough, I have been trying to learn the material on my own b/c my professor has a thick accent and its hard to understand him.
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  6. #6
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    Look at this sequence: a_1 =0 then if n\ge2 define a_n = 2 - a_{n-1}.
    Now you write out the next six terms. What do you see?
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