Neal D. answered • 08/15/16
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NO
arithmetic sequences can be modeled by a linear equation:
Y = dn + a
a= 1st term of seq d=difference between terms
n= term #
geometric sequences can be
modeled by exponential
functions:
Y = ar^n
a = first term, r = ratio
n = # of term
linear and geometric equations will never have the same coordinates
Mark M.
Your example does satisfy the conditions of both. Therefore it is both! Bravo!
Continue questioning the rules on how the rules are made. This shall lead to a deeper understanding.
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08/15/16
Neal D.
Since the terms of a geometric sequence can be modeled by an exponential function,
its equation is similar to that of an exponential function, quite possibly just different
letters:
SEQUENCE: y = arn ; a= 1st term, r =ratio, n = # of term
EXPONENTIAL: y = abx ; a = y-intercept, b = base, x = # of times multiplied
By definition: The b in the exponential function: b > 0 and b ≠ 1
Since the b and the r are basically the same, then the r in the geometric sequence
has the same restrictions as the b in the exponential function,
Therefor: 3, 3, 3, 3, 3, 3,...... would not be an exponential function or a geometric
expression as its r = 1
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08/15/16
Michael W.
Neal, I'm sorry, but I'm not seeing any reliable sources online that say that the common ratio in a geometric sequence can't be 1. I understand that you can use an exponential function to model a geometric sequence, so maybe "b" can't be 1 if you think of it that way...but you don't have to use an exponential function. That's not the formal definition of a geometric sequence.
If I'm missing something, and you have a formal definition of a geometric sequence somewhere that says that "r" can't be 1, then I'm off the mark.
So, to Francis' original question, I think a constant sequence, like 3 3 3 3 3...is both arithmetic and geometric.
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08/16/16
Neal D.
I will get the book out and get the definition!
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08/16/16
Neal D.
All that I see is that the multiplier is the common ratio between any two terms of a geometric sequence, if ratio < 1 then the sequence decreases and if ratio > 1 then the sequence increases.
I believe that some books refer to the geometric sequence: 3, 3, 3, 3, 3, .... as the
TRIVIAL CASE
I have basically exhausted my EXPERT SOURCES on this problem!
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08/16/16
Michael W.
Well, trivial or otherwise, it sounds like having a common ratio of 1 is fine, so a sequence where the same number is repeated over and over again, technically, is a valid geometric sequence.
And, as an arithmetic sequence, the common difference would be zero if we repeat the same number over and over again.
So, there ya go. A weird answer, but it works.
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08/16/16
Michael W.
08/15/16