Which explains why the sequence 81,3,1/9,... is arithmetic or geometric?
I think it's geometric, but I not sure why It either because:
It decreases by 3 or It decreases by 1/27. I think it could be because it decreases 1/27.
Thank you Reiny! you are extremely helpful! Bless you and have a good day!
Is it arithmetic?
Is 3 - 81 = 1/9 - 3 ? , Nope, so it is not arithmetic
Is it geometric?
is 3/81 = (1/9) / 3 ? , Yes, so it is geometric
to be arithmetic, there has to be a common "difference" between consecutive terms
to be geometric, there has to be a common "ratio" between consecutive terms,
which is true in your case, and that common ratio is 1/27
To determine whether the given sequence is arithmetic or geometric, we need to analyze the pattern of the terms.
In an arithmetic sequence, each term is obtained by adding a constant value (called the common difference) to the previous term. So, if the given sequence is arithmetic, we would expect a constant difference between consecutive terms.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant value (called the common ratio). If the given sequence is geometric, we would expect a consistent ratio between consecutive terms.
Let's examine the given sequence: 81, 3, 1/9, ...
To test whether it is an arithmetic sequence, let's compute the difference between consecutive terms:
3 - 81 = -78
1/9 - 3 = -2 8/9
Since the differences are not constant, we can conclude that the sequence is NOT arithmetic.
Now, let's examine whether it is a geometric sequence. To do this, we need to compute the ratio between consecutive terms:
3 / 81 = 1/27
(1/9) / 3 = 1/27
Since both ratios are the same (1/27), we can conclude that the given sequence is, indeed, a geometric sequence with a common ratio of 1/27.
Therefore, your correct intuition is that the sequence is geometric because it decreases by a factor of 1/27 from one term to the next.