the fifth term of an exponential sequence [GP] is greater than the term by 13.5 and the fourth term is greater than the third term by 9 find the common ratio and the first term.

Answer 1

I will assume it says,
"the fifth term of an exponential sequence [GP] is greater than the fourth term by 13.5"

ar^4 - ar^3 = 13.5
ar^3(r - 1) = 13.5

"the fourth term is greater than the third term by 9"
ar^3 - ar^2 = 9
ar^2(r-1) = 9

divide the first equation by the second, the a's cancel, and the r-1 cancels
r = 13.5/9 = 135/90 = 3/2

back in ar^2(r-1) = 9
a(9/4)(1/2) = 9
a = 8

check: the sequence is 8, 12, 18, 27, 40.5,60.75, ....

fourth - third = 27-18 = 9
fifth - fourth = 40.5 - 27 = 13.5
all is good

change the numbers to whatever they should be, stick with the method