2, 1, 1/2, 1/4...
a. arithmetic
b. geometric
c. both
d. neither
Is it d? Thanks!
Or is it b...
b is right. Each number is 1/2 of the previous number.
Thanks you, Ms. Sue. You're sure are a great help.
You are welcome, Chrisley. :-)
To determine whether the sequence 2, 1, 1/2, 1/4... is arithmetic, geometric, both, or neither, we need to understand the differences between these types of sequences.
An arithmetic sequence is one in which the difference between consecutive terms is constant. In other words, if you subtract any term from the next term, you will always get the same value.
A geometric sequence, on the other hand, is one in which the ratio between consecutive terms is constant. If you divide any term by the previous term, you will always get the same value.
Now, let's analyze the given sequence: 2, 1, 1/2, 1/4...
To check if it is an arithmetic sequence, we subtract each term from the next term:
1 - 2 = -1
1/2 - 1 = -1/2
1/4 - 1/2 = -1/4
As you can see, the differences (-1, -1/2, -1/4) are not constant. Therefore, the sequence is not arithmetic.
Next, let's check if it is a geometric sequence by dividing each term by the previous term:
1 / 2 = 1/2
(1/2) / 1 = 1/2
(1/4) / (1/2) = 1/2
The ratios (1/2, 1/2, 1/2) are indeed constant. Therefore, the sequence is geometric.
Since the sequence is geometric but not arithmetic, the answer is:
c. both (geometric)
Actually, all the answers given (a, b, c, d) are wrong.
The correct answer should be
e. CANNOT BE DETERMINED
For a, an arithmetic sequence refers to a sequence that the difference between consecutive terms ((n+1)th term - nth term) is constant. Obviously 1-2 =/= 1/2-1
a is wrong
Thus, c is wrong.
For b, a geometric sequence refers to a sequence that the ratio between consecutive terms i.e. (n+1)th term/nth term is constant. Although the given four terms, such ratio is constant, i.e. 1/2=1/2/1=1/4 / 1/2, the fifth term is not given and you cannot know the general term of the sequence and thus the GENERAL ratio of the sequence.
So, it CANNOT be determined if b is correct if the GENERAL TERM of the sequence is NOT GIVEN and so as d.
Therefore, the answer is e, i.e. CANNOT BE DETERMINED.
(Point to note: You can ONLY determine if a sequence is arithmetic or geometric by knowing its GENERAL TERM and test if the common difference or ratio is CONSTANT.