The fifth term of an exponential sequence (GP) is greater than the fourth term by 13.5 and the fourth term is greater than the third by 9. Find (a) the common ratios (b) the first term.

Answer 1

a r^4 - a r^3 = 13.5
ar^3 - a r^2 = 9
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a (r^4 - r^3) = 13.5
a (r^3 - r^2) = 9
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a = 13.5/(r^4-r^3)
a = 9/(r^3-r^2)
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13.5/(r^4-r^3) = 9/(r^3-r^2)
9 r^4 - 9 r^3 = 13.5 r^3 - 13.5 r^2
9 r^4 - 22.5 r^3 + 13.5 r^2 = 0
r^2 ( 9 r^2 -22.5 r + 13.5) =0
r = 3/2 or r = 1
if r = 1, all terms are the same
if r = 3/2
a (r^3 - r^2) = 9
a r^2 (r-1) = 9
a (9/4)(1/2) = 9
9 a = 9 * 8
a = 8
8 , 12, 18, 27, 40.5 remarkable, it works :)