Geometric sequence

so we have  $ \{  \underbrace{2}_\text{ first term },\underbrace{\frac{1}{5}}_\text{ 2nd term } , \underbrace{...}_\text{ 2 other terms },  \underbrace{a_5}_\text{ 5th term } \}  $

 

the formula to find any term of this geosequence is

$a_n=ar^{n-1}$ where $a_n=n^{th} term$    ;   $ a=\text{first term}$   ;   $r=\text{common ratio}$

 

so far we have $ a_5=2r^{5-1}   $

 

to find the ratio we just write this simple equation $  2\cdot \:r=\frac{1}{5}  \implies  r=\frac{1}{10} $

 

now that we know what the common ratio is, going back and plugging it in the formula we get:

 

$ a_5=2\left(\frac{1}{10}\right)^{5-1}  $

 

$   a_5=2\cdot \frac{1^4}{10^4}  $ 

 

$ a_5=2\cdot \frac{1}{10^4}  $

 

$   a_5=\frac{2}{10^4}  $

 

$ \boxed{a_5=\frac{2}{10000}}  \Leftrightarrow \boxed{ a_5=\frac{1}{5000}}  $

From a1 to a2   is    a1 * r     so r = 1/10

   starting form second term   1/5 * r * r * r   = a5 = 1/5 * 1/1000 = 1/5000