Which of the following statements correctly determines the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, . . .? (1 point) Responses Multiply by the common difference of −3.5. Multiply by the common difference of negative 3.5 . Add the common difference of −3.5. Add the common difference of negative 3.5 . Multiply by the common ratio of −3.5. Multiply by the common ratio of negative 3.5 . Add the common ratio of −3.5. Add the common ratio of negative 3.5 .

Answer 1

To determine the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, we first observe the differences between consecutive terms:

1. 23 - 19.5 = 3.5
2. 19.5 - 16 = 3.5
3. 16 - 12.5 = 3.5

We can see that each term decreases by 3.5.

Therefore, the rule for generating the next term is to add the common difference of -3.5 (which is the same as subtracting 3.5).

The correct responses are:

• Add the common difference of -3.5.

(Note: The options stating "multiply" or "common ratio" are not applicable as this is an arithmetic sequence, not a geometric one.)

Answer 2

What is the common ratio in the following geometric sequence? 2, 12 , 18 , 132 , . . . (1 point) Responses 12 Start Fraction 1 over 2 End Fraction 14 Start Fraction 1 over 4 End Fraction 4 4 −1.5

Answer 3

To determine the common ratio of a geometric sequence, you divide any term by the previous term. Let's calculate the common ratio using the first two terms:

1. First term: \(2\)
2. Second term: \(12\)

The common ratio \(r\) is given by:

\[ r = \frac{\text{second term}}{\text{first term}} = \frac{12}{2} = 6 \]

Now let's check if this ratio holds for the next pair of terms:

3. Third term: \(18\)

Calculating the ratio between the second and third terms:

\[ r = \frac{18}{12} = 1.5 \]

This shows that the sequence is not geometric because the ratios are not the same.

Let's check the ratio for the third and fourth terms:

4. Fourth term: \(132\)

Calculating the ratio between the third and fourth terms:

\[ r = \frac{132}{18} = \frac{132 \div 6}{18 \div 6} = \frac{22}{3} \approx 7.33 \]

The ratios are all different, indicating that this sequence is not a geometric sequence at all. Therefore, there isn't a common ratio.

Based on the options provided (12, \( \frac{1}{2} \), \( \frac{1}{4} \), 4, -1.5), none of these is the correct common ratio, as there is no common ratio due to the inconsistency in the ratios.

If you need further assistance, please clarify if there are more terms or if adjustments can be made to your query.