Question

Which sequences are geometric?

–2.7, –9, –30, –100, ...
–1, 2.5, –6.25, 15.625, ...
9.1, 9.2, 9.3, 9.4, ...
8, 0.8, 0.08, 0.008, ...
4, –4, –12, –20, ...

Answer 1

The theory of geometric sequence. (BRIEFLY)

Suppose you have the following sequence: \( \tiny \\[0.6em] \)
\(\large\color{#0000ff }{ \displaystyle a_1,~~a_2,~~a_3,~~a_4,~~a_5,~~a_6~~\dots }\)

Then, in order for this sequence to be geometric,
it has to yiled the same result in \( \tiny \\[0.6em] \)
\(\large\color{#0000ff }{ \displaystyle a_{n+1}~~/~~a_{n}=r }\)

for all terms of the sequence.

That is,
\(\large\color{#0000ff }{ \displaystyle a_{2}~~/~~a_{1}=r }\)
\(\large\color{#0000ff }{ \displaystyle a_{3}~~/~~a_{2}=r }\)
\(\large\color{#0000ff }{ \displaystyle a_{4}~~/~~a_{3}=r }\)
\(\large\color{#0000ff }{ \displaystyle a_{5}~~/~~a_{4}=r }\)
And so forth...

where each of the above must yiled the same result.

Answer 2

For example
\(\large\color{#0000ff }{ \displaystyle 2,~4,~8,~16 }\)

is a GEOMETRIC sequence because:
\(\large\color{#0000ff }{ \displaystyle 16/8=2 }\)
\(\large\color{#0000ff }{ \displaystyle 8/4=2}\)
\(\large\color{#0000ff }{ \displaystyle 4/2=2 }\)

And notince they all yield the same ratio (r).

Answer 3

And another example
\(\large\color{#0000ff }{ \displaystyle 5,~~25,~~100,~~500, ~~2500 }\)

is NOT a GEOMETRIC sequence, because
\(\large\color{#0000ff }{ \displaystyle 2500/500=5 }\)
\(\large\color{#0000ff }{ \displaystyle 500/100=5 }\)
\(\large\color{#0000ff }{ \displaystyle 100/25=\color{red}{\LARGE 4} }\) (ratio "r" is not same)
\(\large\color{#0000ff }{ \displaystyle 25/5=5 }\)