Question

HELP fast please!!!!!
Classify the sequence as arithmetic, geometric, or neither. If there is not enough information to classify the sequence, choose not enough information.
(12+(-8)+16/3+(-32/9)+64/27) Then, determine if the series is convergent or divergent.

Answer 1

The sequence is geometric. The infinite series does converge.

Note how dividing any given term by the previous one leads to the same result
-8/12 = -2/3
(16/3) divided by -8 = -2/3
and so on...
So the common ratio is r = -2/3. This proves the sequence is geometric.

Since |r| < 1 is true, this means the infinite series converges to some fixed number.

Answer 2

We want to classify the given sequence, we will see that it is a geometric sequence.

So the given sequence is:

12, -8, 16/3, -32/9, 64/27

Because of the sign change, we can conclude that it is not an arithmetic sequence, then we can try to check if it is a geometric sequence.

It will be a geometric sequence if the ratio between two consecutive terms is always the same for any two consecutive terms we take.

Then let's see that:

$\frac{-8}{12} = -0.666...$

$\frac{16/3}{-8} = -0.666...$

$\frac{-32/9}{16/3} = 0.666...$

And so on, so we can see that the ratio is always the same, meaning that this is a geometric sequence.

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