Geometric Progression.
posted by AREMU .
The fifth of an exponential sequence (GP) is greater than the fourth term by 13.5 and fourth term is greater than the thrid by 9.Find:common ratio and the first term.

(x + 9) / x = (x + 22.5) / (x + 9)
x^2 + 18 x + 81 = x^2 + 22.5 x
81 = 4.5 x
a1, a2, 18, 27, 40.5 
ar^4  ar^3 = 13.5
ar^3(r  1) = 13.5 **
ar^3  ar^2 = 9
ar^2(r1) = 9 ***
divide ** by ***
r = 13.5/9 = 1.5
sub into ***
a(1.5)^2 (1.51) = 9
a(1.125) = 9
a = 9/1.125 = 8
first term is 8, common ratio = 1.5 or 3/2
check:
the terms are:
8, 12, 18, 27, 40.5, ...
term5  term4 = 40.527 = 13.5 , good!
term4  term3 = 2718 = 9 , as it should be!
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