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)

Question

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 ratio of −3.5
.
Multiply by the common ratio of  negative 3.5 .

Add the common difference of −3.5
.
Add the common difference of  negative 3.5 .

Multiply by the common difference of −3.5
.
Multiply by the common difference of  negative 3.5 .

Add the common ratio of −3.5
.

GPT-4o mini · Answer

To determine the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, we need to observe what is happening between consecutive terms.

From 23 to 19.5: $ 19.5 - 23 = -3.5 $
From 19.5 to 16: $ 16 - 19.5 = -3.5 $
From 16 to 12.5: $ 12.5 - 16 = -3.5 $

The difference between each consecutive term is -3.5. Therefore, we can see that the rule for generating the next term in this sequence is to add the common difference of -3.5.

So, the correct statement is:

Add the common difference of -3.5.

adaleta · Answer

Classify the following list of numbers as an arithmetic sequence, a geometric sequence, some other sequence, or not a sequence.

9, 4, −1
, −6
, . . .

(1 point)
Responses

some other sequence
some other sequence

arithmetic sequence
arithmetic sequence

not a sequence
not a sequence

geometric sequence

GPT-4o mini · Answer

To classify the sequence $ 9, 4, -1, -6, \ldots $, we need to check if it is an arithmetic sequence, a geometric sequence, or something else. 1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant. - From $ 9 $ to $ 4 $: $ 4 - 9 = -5 $ - From $ 4 $ to $ -1 $: $ -1 - 4 = -5 $ - From $ -1 $ to $ -6 $: $ -6 - (-1) = -5 $ The common difference is $-5$, which is consistent between all terms. Therefore, this sequence is an **arithmetic sequence**. 2. **Geometric Sequence**: In a geometric sequence, the ratio between consecutive terms is constant. - From $ 9 $ to $ 4 $: $ \frac{4}{9} $ - From $ 4 $ to $ -1 $: $ \frac{-1}{4} $ - From $ -1 $ to $ -6 $: $ \frac{-6}{-1} = 6 $ The ratios are not constant, so it is not a geometric sequence. Based on this analysis, the correct classification for the sequence $ 9, 4, -1, -6, \ldots $ is: **Arithmetic sequence.**

adaleta · Answer

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

GPT-4o mini · Answer

To find the common ratio in a geometric sequence, we divide any term by the previous term. Let's calculate the ratios using the terms given in the sequence:

From 2 to 12: \[ \text{Ratio} = \frac{12}{2} = 6 \]
From 12 to 18: \[ \text{Ratio} = \frac{18}{12} = 1.5 \]
From 18 to 132: \[ \text{Ratio} = \frac{132}{18} = 7.33 (approximately) \]

Since the ratios between the consecutive terms are not the same, it indicates that this is not a geometric sequence.

However, if we check the options provided, it seems there might be confusion in how the terms are interpreted.

To correct and address the computations for clarity, we must ensure the terms given actually form a geometric sequence. Let's carefully check:

Under a geometric sequence definition, the common ratio must be constant across all terms.

As we can observe here, the ratios vary, which implies the sequence is not geometric.

If you want to confirm if there is a specific fraction that results in a particular product at some steps, please provide clarification on how the sequence is defined, or check if the sequence presented has any inconsistencies.

Conclusion: Given the numbers provided, none of the possible answers represent the common ratio of a geometric sequence, because the ratios are not constant.

adaleta · Answer

so the answer?

GPT-4o mini · Answer

I apologize for the confusion in my earlier response. Upon reevaluation, let's clarify:

You gave the sequence: 2, 12, 18, 132, ...

To determine the common ratio in what might be intended as a geometric sequence, we'll check the ratios again:

From 2 to 12: \[ \frac{12}{2} = 6 \]
From 12 to 18: \[ \frac{18}{12} = 1.5 \]
From 18 to 132: \[ \frac{132}{18} = 7.33 , (\text{approximately}) \]

Since the ratios are not consistent, this sequence does not form a geometric sequence, and thus it does not have a common ratio.

None of the provided response options (14, 1/4, 4, -1.5, 12) correctly describe a common ratio because this is not an actual geometric sequence based on the numbers provided.

If I were to choose the best way to represent that this sequence does not have a constant common ratio or accurately follows geometric sequence rules, I would have to state that it doesn't fit the definitions required for a single common ratio to be appropriate.

If the purpose is to find a significant common ratio from the first two elements (assuming other terms were misstated), the ratio you would highlight is 6 from $2$ to $12$. However, please note that this is not consistent across other pairs.

In short, based solely on the provided options, the sequence does not fit any geometric pattern correctly. If you still need an answer from the provided option set or something else clarified, please provide further context!

adaleta · Answer

What is the common ratio in the following geometric sequence? 2, 1/2 , 1/8 , 1/32 , . . . (1 point) Responses 1/4 Start Fraction 1 over 4 End Fraction 4 4 −1.5 negative 1.5 1/2

GPT-4o mini · Answer

To find the common ratio in the geometric sequence $2, \frac{1}{2}, \frac{1}{8}, \frac{1}{32}, \ldots$, we divide any term by the term that comes before it.

Let's calculate the ratio:

From $2$ to $\frac{1}{2}$: \[ \text{Common Ratio} = \frac{\frac{1}{2}}{2} = \frac{1}{4} \]
From $\frac{1}{2}$ to $\frac{1}{8}$: \[ \text{Common Ratio} = \frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{8} \times \frac{2}{1} = \frac{2}{8} = \frac{1}{4} \]
From $\frac{1}{8}$ to $\frac{1}{32}$: \[ \text{Common Ratio} = \frac{\frac{1}{32}}{\frac{1}{8}} = \frac{1}{32} \times \frac{8}{1} = \frac{8}{32} = \frac{1}{4} \]

In all cases, the common ratio is $\frac{1}{4}$.

Thus, the correct answer is:

$\frac{1}{4}$ (or "Start Fraction 1 over 4 End Fraction").

adaleta · Answer

Given the first five terms of the sequence 19, 15, 11, 7, 3 . . .
, use subscript notation to describe a7
.(1 point)
Responses

a7=−4
a subscript 7 baseline equals negative 4

a7=−9
a subscript 7 baseline equals negative 9

a7=−1
a subscript 7 baseline equals negative 1

a7=−5
a subscript 7 baseline equals negative 5

GPT-4o mini · Answer

To find $ a_7 $ in the sequence $ 19, 15, 11, 7, 3, \ldots $, we first observe that this is an arithmetic sequence. The common difference can be calculated by subtracting the consecutive terms: - From $ 19 $ to $ 15 $: $ 15 - 19 = -4 $ - From $ 15 $ to $ 11 $: $ 11 - 15 = -4 $ - From $ 11 $ to $ 7 $: $ 7 - 11 = -4 $ - From $ 7 $ to $ 3 $: $ 3 - 7 = -4 $ The common difference $ d $ is $-4$. Now we can use the formula for the $ n $-th term of an arithmetic sequence: $$ a_n = a_1 + (n - 1) \cdot d $$ Where: - $ a_1 = 19 $ (the first term) - $ d = -4 $ - $ n = 7 $ Substituting in the values: $$ a_7 = 19 + (7 - 1) \cdot (-4) $$ $$ a_7 = 19 + 6 \cdot (-4) $$ $$ a_7 = 19 - 24 $$ $$ a_7 = -5 $$ Thus, the correct notation for $ a_7 $ is: **$ a_7 = -5 $** (or "a subscript 7 baseline equals negative 5").