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Math Help - Sequence Forumla?

  1. #1
    Member looi76's Avatar
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    Sequence Forumla?

    Suggest inductive definitions which would produce the following sequences:
    1) 2, 4, 6, 8, 10 ...
    2) \frac{1}{3},\frac{1}{8}, \frac{1}{27}, \frac{1}{81}

    Whats the formula to produce the definition?
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  2. #2
    Super Member Aryth's Avatar
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    Well, let's just look at them individually:

    1) 2, 4, 6, 8, 10, ...

    It's simple to find what produced this sequence, just take a look at it:

    (2*1), (2*2), (2*3), (2*4), (2*5), ...

    It looks like the sequence is defined by:

    \color{blue}a_n = 2n

    2) \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, ...

    Let's take a closer look:

    \frac{1}{3^1}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, ...

    It looks like this one is defined by:

    \color{red}a_n = \frac{1}{3^n}

    And there you go.
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  3. #3
    Member looi76's Avatar
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    That was easy.. how can I solve these:
    1) b-2c, b-c, b+c, ...
    2) 1, 1+x, (1+x)^2, (1+x)^3, ...
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  4. #4
    Super Member Aryth's Avatar
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    1)

    \sum_{k=-2}^{n}b+kc

    or

    a_n = b + nc

    2)

    a_n = (1 + x)^n

    or

    \sum_{k=0}^{n}(1 + x)^k
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