Answer
The correct option is $I$.
Work Step by Step
(F) $1,-1,1,-1....$
$a_1=1$
$a_2=-1$
Common difference $d=a_2-a_1=-1-1=-2$
$a_3=a_2+d=-1-2=-3$
Not an arithmetic sequence.
(G) $16,24,36,54....$
$a_1=16$
$a_2=24$
Common difference $d=a_2-a_1=24-16=8$
$a_3=a_2+d=24+8=32$
Not an arithmetic sequence.
(H) $1,4,9,16....$
$a_1=1$
$a_2=4$
Common difference $d=a_2-a_1=4-1=3$
$a_3=a_2+d=4+3=7$
Not an arithmetic sequence.
(I) $5,5,5,5....$
$a_1=5$
$a_2=5$
Common difference $d=a_2-a_1=5-5=0$
$a_3=a_2+d=5-0=5$
Not an arithmetic sequence.
Now check for geometric sequence.
$a_1=5$
$a_2=5$
Common ratio $r=\frac{a_2}{a_1}=\frac{5}{5}=1$
$a_3=a_2\cdot r=5\cdot 1=5$
This is a geometric sequence.
