## are any sequences both arithmetic and geometric?

Sequences, counting (including probability), logic and truth tables, algorithms, number theory, set theory, etc.
odysseus
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Joined: Sat Jul 25, 2009 3:07 pm

### are any sequences both arithmetic and geometric?

Are there any sequences which are both arithmetic and geometric? I don't think so, but how would I prove this?

stapel_eliz
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Joined: Mon Dec 08, 2008 4:22 pm
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...how would I prove this?
You'd probably need to start with the formulas for each type of sequence.

For an arithmetic sequence, the terms are a, a + d, a + 2d, a + 3d, etc.

For a geometric sequence, the terms are a, ar, ar2, ar3, etc.

Arithmetic sequences, after some point, will always be the same sign. (They can change sign, as -3, -1, 1, 3, 5, etc, but will eventually, due to their additive nature, always stay the changed sign forever after.) So let's assume, for simplicity, that the sign is always the same.

Geometric sequences will be always the same sign, or else will alternate signs if the common ratio "r" is negative. For the geometric sequence to be also arithmetic, clearly r will have to be positive. So we'll need a geometric sequence which is always the same sign.

Assuming that some geometric sequence is also arithmetic, we get a common difference for the terms of the geometric sequence:

. . . . .$ar\, -\, a\, =\, d$

. . . . .$ar^2\, -\, ar\, =\, d$

Then:

. . . . .$ar^2\, -\, ar\, =\, ar\, -\, a$

. . . . .$ar^2\, -\, 2ar\, +\, a\, =\, 0$

. . . . .$r\, =\, \frac{2a\, \pm\, \sqrt{4a^2\, -\, 4a^2}}{2a}\, =\, \frac{2a\, \pm\, 0}{2a}\, =\, 1$

Now go review the definition of the common ratio for a geometric sequence. Is the above allowed?