is there any way to prove that 0,0,0,0,... is the only sequence that can have 2 different "r" values?

Question

i tried to understand one but it made no sense and it looked like it skipped some steps. i really want to understand the logic

Answer 1

I have already answered this question here:
http://answers.yahoo.com/question/?qid=20070904160409AAA6ilf

Here is the proof, perhaps in simpler, less mathematical terms.

Take two geometric sequences, with different "ratios."

They must have the first initial value, call that value n.

The first way to write the sequence is:
n, rn, r²n, r³n, ...
The second way is:
n, sn, s²n, s³n, ...

If that's the same sequence, with two different ratios (r & s), then we know the second term in each is equal:

rn = sn

rn - sn = 0

(r-s)n = 0

Because s ≠ r, we know r-s ≠ 0. That means we can divide by (r-s). We can divide by anything except 0.

So

n = 0 / (r-s) = 0

If the first term of the sequences is 0, then all other terms are 0, since:


0 = 0
0×r = 0
0×r² = 0
etc...

Answer 2

I'm assuming that you meant to ask if it is the only geomtric sequence that can have two different r values.

In a geometric sequence, a_(n+1) = r * a_n
(i'm using _ to represent subscripts here)
Suppose there are two possible r, r_1 and r_2, then
a_(n+1) = r_1 * a_n and
a_(n+1) = r _2 * a_n
subtract the second equation from the first.
0 = (r_1 - r_2) * a_n
for a_n not equal to zero we can divide by a_n on both sides.
0 = r_1 - r_2
r_2 = r_1.
Which shows that there can be only one r unless a_n = 0

Answer 3

uhhh

Answer 4

there is if its set up as an array