Question

The third term of a sequence is 120. The fifth term is 76.8. Write an explicit rule representing this geometric sequence.

Answer 1

The nth term of the geometric sequence is:
an=ar^(n-1)
where 
a=first term
r=common ratio
n=nth term
from the question:
120=ar(3-1)
120=ar^2
a=120/(r^2)....i
also:
76.8=ar^(5-1)
76.8=ar^4
a=76.8/r^4.....i
thus from i and ii
120/r^2=76.8/r^4
from above we can have:
120=76.8/r²
120r²=76.8
r²=76.8/120
r²=0.64
r=√0.64
r=0.8
hence:
a=120/(0.64)=187.5
therefore the formula for the series will be:
an=187.5r^0.8

Answer 2

The nth term of the geometric sequence is:
 an=ar^(n-1)
 where
a=first term
 r=common ratio
 n=nth term
 from the question:
 120=ar(3-1)
 120=ar^2
 a=120/(r^2)....i
 also:
 76.8=ar^(5-1)
 76.8=ar^4
 a=76.8/r^4.....i
  thus from i and ii
 120/r^2=76.8/r^4
 from above we can have:
 120=76.8/r²
 120r²=76.8
 r²=76.8/120
 r²=0.64
 r=âš0.64
 r=0.8
 hence: a=120/(0.64)=187.5 therefore the formula for the series will be: an=187.5r^0.8