tcesni
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can a sequence be arithmatic and geometric??
can a sequence be arithmatic an geometric??
can a sequence be arithmatic an geometric??
Can a sequence be arithmatic and geometric?
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Write out the formula for the nth term of each and see if they can be equal for all n for some values for d and r:tcesni said:
[tex]a_n = a_1 + d(n-1)[/tex]
[tex]a_n = a_1 r^{n-1}[/tex]
AM
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Yes, they definitely can be equal.
Gold Member
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JasonRox said:
Did you find one yet?
GoldPheonix
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Well, two sequences can equal each other at certain points, but a sequence cannot be both geometric and arithmetic. It be a combination of both of them, no doubt, but then it is not arithemetic nor geometric from my understanding of sequences.
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Sure they can.
The general n'th term in an aritmetic sequence is
[tex]a_{n}=a_{0}r^{n}[/tex]
whereas the general term in a geometric sequence is:
[tex]g_{n}=g_{0}+kn[/tex]
where [itex]a_{0},g_{0},r,k[/itex] are constants independent of n.
A sequence that is both arithmetic and geometric fulfills BOTH conditions for all choices of n, which means that we must have:
[tex]g_{0}+kn=a_{0}r^{n} (*)[/tex]
That is, we must have [itex]g_{n}=a_{n}[/itex] for all n.
Now, you might fiddle about to find constants [itex]a_{0}, g_{0},k,r[/itex]
so that (*) holds for all n. It can be done.
The general n'th term in an aritmetic sequence is
[tex]a_{n}=a_{0}r^{n}[/tex]
whereas the general term in a geometric sequence is:
[tex]g_{n}=g_{0}+kn[/tex]
where [itex]a_{0},g_{0},r,k[/itex] are constants independent of n.
A sequence that is both arithmetic and geometric fulfills BOTH conditions for all choices of n, which means that we must have:
[tex]g_{0}+kn=a_{0}r^{n} (*)[/tex]
That is, we must have [itex]g_{n}=a_{n}[/itex] for all n.
Now, you might fiddle about to find constants [itex]a_{0}, g_{0},k,r[/itex]
so that (*) holds for all n. It can be done.
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Uh, Arildno- you have the definitions of "geometric" and "arithmetic" sequences reversed.
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The trivial example is 1, 1, 1, 1, which is both arithmetic with respect to 0 (I don't think that's proper terminology, but I'll be damned if it doesn't sound good) and geometric with respect to 1
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Uhmm, blarrg, huge embarassment.HallsofIvy said:
Please tell me when I can take my head out of the bucket.
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Actually, the sequence a, a, a, a, ..., for any a, is both arithmetic (a+ 0n) and geometric (a(1n)). It's easy to show that any sequence that is both arithmetic and geometric must be of the form a, a, a, a, ... for some a.
(You may now remove head from bucket, arildno.)
(You may now remove head from bucket, arildno.)