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Thread: Exponential Progression

  1. #1 Exponential Progression 
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    It seems exponential progression and geometric progression both mean the same thing in common language.

    But geometric progression is getting to the next term by multiplication of a constant "common ratio".

    For example: 2, 4, 8, 16, 32... (a1 = 2, common ratio r​ = 2)

    Of course the partial sums increase at a non-linear rate.

    Yes this might be non-linear, but for me, it doesn't really fit "exponential" that well because it works by multiplication.

    What if we had progression that worked up the next level of operations, exponentiation, as in powers?

    Here, a term is taken to a constant "common power" to produce the next term.

    Like this: 2, 4, 16, 256, 65536... (a1 = 2, common power p = 2)

    So could we apply this to the real world?


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  3. #2  
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    Quote Originally Posted by Eonos115 View Post
    So could we apply this to the real world?
    Also, is there a possible explicit rule to find the nth term? The recursive is obvious though. And also a rule for partial sums?
    I'm guessing for these rules to be short as possible would require tetration notation, as in 4n, which is n^n^n^n.


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    Forum Professor wallaby's Avatar
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    well you've already done some of the grunt work yourself. you've described a sequence governed by the rule, where 'r' denotes some constant power. Note that we can simplify this further, . Partial sums are an easy extension, . For convergence of the partial sums we require (assuming i didn't mess up when figuring out how to subtract tetration terms)
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    Okay. Thank you. I have no idea how to algebraically work with any of this...

    All the equations seem correct.

    Using that method we can define tetrative progression.

    So for the recursive rule:

    Attachment 180

    But that's good enough considering how a simple sequence like:

    2 (--> 2^2 -->) 4 (--> 4^4 -->) 256 (--> 256^256 -->) ... (t = 2)

    So by the fourth term you're using scientific notation to a large scale, and after that it would be unimaginably enormous.
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    Quote Originally Posted by Eonos115 View Post
    Attachment 180
    .
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  7. #6  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by Eonos115 View Post
    So could we apply this to the real world?
    Are you suggesting trying to get the mainstream media, for example, to use the correct meanings of these words?

    Let us know how that works out for you
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  8. #7  
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    Quote Originally Posted by Strange View Post
    Quote Originally Posted by Eonos115 View Post
    So could we apply this to the real world?
    Are you suggesting trying to get the mainstream media, for example, to use the correct meanings of these words?

    Let us know how that works out for you
    No I meant what real-world situations involve an exponential progression?

    A cell dividing into 2, then those 2 each into 2, you have 4, and so on, and it's constant geometric progression.

    But could we apply an exponential sequence?
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