The fifth term of an exponential sequence is greater than the fourth term by 13.5 and the fourth term is greater than the third by 9. Find: a) the common ratio b) the common difference
The question says we are dealing with an exponential sequence, so there
is a common ratio. To ask for a common difference makes no sense.
Nevertheless, the AI gives us some gobble-de-goop about common difference
from the given:
ar^4 - ar^3 = 13.5 ----> ar^3(r-1) = 13.5
ar^3 - ar^2 = 9 -----> ar^2(r-1) = 9
divide them
r = 13.5/9 = (135/10)(1/9) = 3/2
a) The common ratio is the ratio of consecutive terms in the sequence. In this case, the fourth term is greater than the third by 9, so the common ratio is 9/13.5 = 0.666666667.
b) The common difference is the difference between consecutive terms in the sequence. In this case, the fifth term is greater than the fourth by 13.5, so the common difference is 13.5.