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