Tag: understanding geometric progressions

If a sequence is a finite geometric sequence, then :

$1 + 0.5 + 0.25 + 0.125 $$ is a finite geometric progression.
Here the common ratio is $0.5$.
An finite geometric series is the sum of an finite geometric sequence.

An infinite geometric series is the sum of an infinite geometric sequence.
So, $1 + 2 + 4 + 8 +....$ is an infinite geometric sequence.
Here the common ratio is $2$ and it is never ending.

Given:

This is G.P in which  $a=2,r=\cfrac{{2}^{2}}{2}=2$ and $n=9$
$\therefore$ ${S} _{n}=\cfrac{a({r}^{n}-1)}{(r-1)}=\cfrac{2\times ({2}^{9}-1)}{(2-1)}=2\times (512-1)=2\times 511=1022$.

Here $a=3$ and $r=\cfrac{6}{3}=2$. Let the number of terms be $n$$.
Then, ${t}_{n}=384$ $\Rightarrow$ $a{r}^{n-1}=384$
$\Rightarrow$ $3\times {2}^{n-1}=384$
$\Rightarrow$ ${2}^{n-1}=128={2}^{7}$
$\Rightarrow$ $n-1=7$
$\Rightarrow$ $n=8$
$\therefore$ Number of terms $=8$.

1. Consider the two numbers $1$ and $4$, geometric mean of $1$ and $4$ is $\sqrt {1 \times 4} = 2$.
When each number is reduced by $2$, the numbers become $-1$ and $2$ whose geometric mean does not exist.
2. Now consider two numbers $2$ and $7$. Their geometric mean is $\sqrt {14}$. The new numbers are $4$ and $9$ whose geometric mean is $\sqrt {4\times 9} = 6$ which is not equal to $2\sqrt {14}$.
Thus only statement $1$ is true.

We are given $a,b,c$ are in G.P.

The common ratio between two numbers has to be non-zero for it to form a G.P.

Difference between consecutive terms is $(4-2), (8-4), (16-8)$ and so on, i.e. 2,4,8 and so on.
Since the difference is not constant, it is not an AP.