Find the missing terms of the following geometric sequence. Assume all terms are positive. ​(​Hint: The geometric mean of the first and fifth terms is the third​ term.) 2.5, ___, ___, ___, 202.5, ...

Answer 1

Know a_n+1=r*a_n

where r is the ratio between terms

a_n+2=r*a_n+1=r²a_n

...

a_n+4=r⁴*a_n

here a_n=2.5 and a_n+4=202.5

r=+/-(202.5/2.5)^(1/4)

corrected to remove negative entries

r=+/-3

problem specifies positive entries only, so r=+3

a_n+1=7.5

a_n+2=22.5

a_n+3=67.5

Answer 2

a1 =2.5

a5 = 202.5

an=a1r^(n-1)

a5 = 2.5(r^4)

202.5=2.5r^4

r^4 = 202.5/2.5 = 405/5= 81

r=4th root of 81=sqr9 = 3

a2= 2.5 x 3 = 7.5

a3 = 7.5 x 3 = 22.5

a4 = 22.5 x 3 = 67.5

a5 = 67.5 x 3 = 202.5

2.5, 7.5, 22.5, 67.5, 202.5, ...

or

2.5, -7.5, 22.5, -67.5, 202.5, since sqr9 = + or -3 , but the problem assumes all terms are >0, so go with the 1st solution with r=+3

geometric mean of a1 and a5 = sqr(a1a5)

a1=2.5, a5 =202.5

sqr(2.5x202.5) = sqr506.75) = 22.5= a3

actually 4th root of 81 has 4 roots, 3, -3 and two more roots

r^4 = 81

r4-81 = 0

(r^2 -9)(r^2+9) =0

r^2 = 9

r =+/-3 are two roots

also two imaginary roots

r^2 = -9

r =+/-3i

then

the geometric sequence could be

2.5, -7.5i, 22.5, -67.5i, 202.5, ... which still has the geometric mean of a1 and a5 = a3

if r=3i

2.5, 7.5i, -22.5, 67.5i, 202.5, ... doesn't have a3 = sqr((2.5)(202.5)), it has a3 = -sqr((2.5)(202.5))

the geometric mean only applies to positive numbers