Sequences and series

===========================================================

Explanation:

The sequence is not arithmetic because the distance between adjacent terms keeps increasing. For instance, the gap from 0.4 to 1.6 is 1.6-0.4 = 1.2, while the gap from 1.6 to 6.4 is 6.4-1.6 = 4.8. There is no common difference, so we can rule out choices A and B.

To check if we have a geometric sequence, divide each term by its previous term

(term2)/(term1) = (1.6)/(0.4) = 4

(term3)/(term2) = (6.4)/(1.6) = 4

(term4)/(term3) = (25.6)/(6.4) = 4

Each time we get 4 as a result, so this sequence is geometric and the common ratio is r = 4.

Put another way: each new term is found by multiplying the previous term by 4.

0.4*4 = 1.6

1.6*4 = 6.4

6.4*4 = 25.6

This all leads to D being the answer.

Side note: the nth term of this geometric sequence is where n is a positive integer and starts at n = 1.