Tag: geometric sequences

$a, ar, ar^2, ar^3, ar^4....$ is an infinite geometric progression.

Given sequence is $a,ar,ar^2, ar^3, ar^4$.

$1 + 0.5 + 0.25 + 0.125....$ is an example of infinite geometric progression.
Here the common ratio is $0.5$.
An infinite geometric series is the sum of an infinite geometric progression.

An geometric sequence containing infinite number of terms. It has a common ratio which is same throughout.
Example: $1 + 0.5 + 0.25 + 0.125....$ is an infinite geometric sequence.
Here the common ratio is $0.5$.

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 $=8$.