9. Is the sequence geometric? If so, identify the common ratio.
1/4 , 3/16 , 9/64 , 27/256 , 81/1,024
yes; 1/3
yes;3/4
not geometric
yes; 2/9
divide each term by its previous one to get 3/4 each time, so .....
its 3/4
To determine if a sequence is geometric, we need to check if there is a fixed ratio between consecutive terms. In this case, let's check if there is a common ratio between the terms:
To find the common ratio, divide each term by the preceding term:
(3/16) / (1/4) = (3/16) * (4/1) = 12/16 = 3/4
(9/64) / (3/16) = (9/64) * (16/3) = 144/192 = 3/4
(27/256) / (9/64) = (27/256) * (64/9) = 1728/2304 = 3/4
(81/1024) / (27/256) = (81/1024) * (256/27) = 20736/27648 = 3/4
Since the ratios between all consecutive terms are the same, namely 3/4, the sequence is geometric. Therefore, the common ratio is 3/4.