Answer
$S_{n}=\dfrac {1\times \left( 1-\left( 0.9\right) ^{5}\right) }{1-0.9}=4.0951$
Work Step by Step
if it is arithmetic sequence then
$a_{n+1}+a_{n-1}=2a_{n}$
if it is geometric sequence then
$a_{n+1}\times a_{n-1}=a^{2}_{n}$
$1+\left( 0.9\right) ^{2}=1.81\neq 2\times 0.9$ so this is not arithmetic sequence
$\left( 0.9\right) ^{2}\times 1=\left( 0.9\right) ^{2}$ so this is geometric sequence
Sum of geometric sequence is
$S_{n}=\dfrac {a\left( 1-r^{n}\right) }{1-r};r=\dfrac {0.9}{1}=0.9;n=5;a=1$
So we get
$S_{n}=\dfrac {1\times \left( 1-\left( 0.9\right) ^{5}\right) }{1-0.9}=4.0951$
