Can a sequence be arithmatic and geometric?

In summary, two sequences can equal each other at certain points, but a sequence cannot be both geometric and arithmetic.
  • #1
tcesni
3
0
can a sequence be arithmatic and geometric??

can a sequence be arithmatic an geometric??
 
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  • #2
tcesni said:
can a sequence be arithmatic an geometric??
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:

[tex]a_n = a_1 + d(n-1)[/tex]

[tex]a_n = a_1 r^{n-1}[/tex]

AM
 
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  • #3
Yes, they definitely can be equal.
 
  • #4
JasonRox said:
Yes, they definitely can be equal.

Did you find one yet?
 
  • #5
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.
 
  • #6
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. :smile:
 
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  • #7
Uh, Arildno- you have the definitions of "geometric" and "arithmetic" sequences reversed.
 
  • #8
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
 
  • #9
HallsofIvy said:
Uh, Arildno- you have the definitions of "geometric" and "arithmetic" sequences reversed.
Uhmm, blarrg, huge embarassment.
Please tell me when I can take my head out of the bucket. :redface:
 
  • #10
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.)
 

Related to Can a sequence be arithmatic and geometric?

1. Can a sequence be both arithmetic and geometric?

Yes, a sequence can be both arithmetic and geometric. In an arithmetic sequence, each term is obtained by adding a fixed number to the previous term. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number. It is possible for a sequence to exhibit both of these patterns, making it both arithmetic and geometric.

2. Are there any examples of sequences that are both arithmetic and geometric?

Yes, there are many examples of sequences that are both arithmetic and geometric. One example is the sequence 1, 2, 4, 8, 16, 32, ... where each term is obtained by doubling the previous term. This sequence is both arithmetic (with a common difference of 1) and geometric (with a common ratio of 2).

3. How can you tell if a sequence is both arithmetic and geometric?

To determine if a sequence is both arithmetic and geometric, look for a pattern in the terms. If each term can be obtained by both adding a fixed number and multiplying a fixed number to the previous term, then the sequence is both arithmetic and geometric.

4. Can a sequence be both arithmetic and geometric if the terms are not whole numbers?

Yes, a sequence can be both arithmetic and geometric even if the terms are not whole numbers. The common difference or common ratio can be a decimal or fraction, as long as it is a fixed value that is added or multiplied to the previous term.

5. Is there a specific formula for calculating a sequence that is both arithmetic and geometric?

No, there is not a specific formula for calculating a sequence that is both arithmetic and geometric. However, if you know the pattern of the sequence, you can use it to determine any term in the sequence. For example, if a sequence is both arithmetic and geometric, you can use the formula for an arithmetic or geometric sequence to find any term.

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