Why $b^2=a\cdot c$ is being used when the terms are not consecutive?

Question

Okay most of us know that if $a,b,c$ are three terms of a sequence then they are in G.P. if $b^2=a\cdot c$. But isnt this formula $b^2=a\cdot c$ is only applicable when the terms are consecutive. Please tell me why is it being used in solving the questions when the term are not consecutive.

Answer 1

If $a$, $b$ and $c$ the nth, (n+k)th and (n+2k)th terms respectively (i.e. equidistant terms) in a GP, then it means you have to multiply $a$ by $r$ $k$ times (i.e. by $r^k$) to get to $b$. Similarly, you have to furthur multiply $b$ by $r$ $k$ times to get to $c$. Hence, $a$, $b$, and $c$ are the consecutive terms of another GP with common ratio $r^k$.

Answer 2

If we have a geometric sequence such that $a_n = ar^{n-1}$, then we have

$$a_{n+m}^2 = a_n \cdot a_{n+2m}$$

since $$\frac{a_{n+m}}{a_n}=r^m = \frac{a_{n+2m}}{a_{n+m}}.$$

They form a new geometric progression.

Answer 3

In a GP the term next to one is just multiplied by a definite ratio. This means every term has a common multiplied term that you may say as $r$.

The fact that every next term is defined as $\text{previous-term}\cdot r$ in a GP.

So if $a_n = x$
Then $a_k = xr^k$
and $a_l= xr^l$

This you may see holds $b^2=a\cdot c$, if you take $a_n$ as $a$ and others as $b$ and $c$ respectively.

UPDATE: This relation holds for all times when the terms are equally spaced in a GP.

Because of the fact that equally spaced terms in a GP, have equal powers of $r$ multiplied to them.

Answer 4

Because, the formula applies to any indices in arithmetic progression. Consecuitive indices just happen to form an arithmetic progression with distance 1.