Find all sequences which are simultaneously an Arithmetic Progression and a Geometric Progression.

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- Jul 14th 2008, 06:27 AMfardeen_genQuestion on finite series?
Find all sequences which are simultaneously an Arithmetic Progression and a Geometric Progression.

- Jul 14th 2008, 06:34 AMkalagota
--- Arithmetic prog.

--- Geometric prog.

from geometric prog.. we have

so, we have - Jul 14th 2008, 06:44 AMfardeen_genQuote:

a(subscript n) = a(subscript 1) + (n - 1)d, isn't it? - Jul 14th 2008, 06:49 AMkalagota
oh yes.. just a typo error..

- Jul 14th 2008, 07:52 AMflyingsquirrel
Hi

Let be such a sequence. There exists two real numbers and such that for any non-negative integer

Substitute (2) in (1) :

- If , hence is a constant sequence. This implies that the common difference has to be hence for any non-negative integer .
- If , hence for any non-negative integer and is a constant sequence.

At this point we've shown that if a progression is both arithmetic and geometric than the sequence is constant. Now, one has to show that if a sequence is constant then it is both arithmetic and geometric : it will allow us to claim that "A sequence is both arithmetic and geometric iff it is constant". - If , hence is a constant sequence. This implies that the common difference has to be hence for any non-negative integer .