Is the sequence 1.6, 0.8, 0.4, 0.2,... arithmetic, geometric, or neither? Explain your answer.
To determine if the sequence 1.6, 0.8, 0.4, 0.2,... is arithmetic, geometric, or neither, we need to examine the differences between consecutive terms.
Let's find the differences:
First difference:
0.8 - 1.6 = -0.8
Second difference:
0.4 - 0.8 = -0.4
Third difference:
0.2 - 0.4 = -0.2
From the differences, we can see that they are all negative and getting smaller in magnitude. This indicates that the sequence is decreasing.
Now, let's check if the ratios between consecutive terms are constant:
First ratio:
0.8 / 1.6 = 0.5
Second ratio:
0.4 / 0.8 = 0.5
Third ratio:
0.2 / 0.4 = 0.5
The ratios between consecutive terms are all equal to 0.5, which is a constant value.
From the analysis above, we can conclude that the sequence 1.6, 0.8, 0.4, 0.2,... is a geometric sequence because the ratios between consecutive terms are constant. The common ratio, in this case, is 0.5.
To determine whether the given sequence 1.6, 0.8, 0.4, 0.2,... is arithmetic, geometric, or neither, we need to identify the pattern in the sequence.
For an arithmetic sequence, the difference between consecutive terms remains constant. In other words, each term is obtained by adding the same value to the previous term.
For a geometric sequence, each term is obtained by multiplying the previous term by a fixed value called the common ratio.
Let's examine the given sequence:
1.6, 0.8, 0.4, 0.2,...
To determine if it's an arithmetic sequence, we can subtract each term from the next term:
0.8 - 1.6 = -0.8
0.4 - 0.8 = -0.4
0.2 - 0.4 = -0.2
As you can see, the differences between consecutive terms are not constant. In an arithmetic sequence, the differences would have been the same value each time. Therefore, we can conclude that the given sequence is not arithmetic.
Next, let's check if it's a geometric sequence by dividing each term by the previous term:
0.8 / 1.6 = 0.5
0.4 / 0.8 = 0.5
0.2 / 0.4 = 0.5
The quotients are all equal to 0.5, which means the ratio between consecutive terms is constant. Therefore, the given sequence is a geometric sequence with a common ratio of 0.5.
In conclusion, the sequence 1.6, 0.8, 0.4, 0.2,... is a geometric sequence.