Responses
Add the common difference of −3.5
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Add the common difference of negative 3.5 .
Multiply by the common difference of −3.5
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Multiply by the common difference of negative 3.5 .
Add the common ratio of −3.5
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Add the common ratio of negative 3.5 .
Multiply by the common ratio of −3.5
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11 answers
The sequence provided is 23, 19.5, 16, 12.5, ...
To determine the next term in the sequence, we can observe how each term relates to the previous one:
Thus, each term decreases by 3.5.
So the correct statement for generating the next term in the sequence is:
Add the common difference of −3.5 .
To determine the classification of the sequence \(9, 4, -1, -6, \ldots\), we should check whether it is an arithmetic sequence, a geometric sequence, or some other type of sequence.
Arithmetic Sequence: An arithmetic sequence has a constant difference between consecutive terms.
Since the difference between each pair of consecutive terms is constant at \(-5\), this is an arithmetic sequence.
Geometric Sequence: A geometric sequence has a constant ratio between consecutive terms. We can check:
As the ratios are not constant, it is not a geometric sequence.
Based on this analysis, the given numbers form an arithmetic sequence.
Final Answer: Arithmetic sequence.
To find the common ratio in a geometric sequence, we divide any term by the preceding term. Let's calculate the ratios using the terms provided in the sequence \(2, 12, 18, 132, \ldots\):
From the first term to the second term: \[ r = \frac{12}{2} = 6 \]
From the second term to the third term: \[ r = \frac{18}{12} = 1.5 \] (Notice: This does not match the previous ratio, indicating it may not be a geometric sequence).
From the third term to the fourth term: \[ r = \frac{132}{18} = \frac{132}{18} \approx 7.33 \quad (\text{which is also not a constant ratio}) \]
Since we find that the ratios between consecutive terms do not remain constant, it appears that this is not a geometric sequence. However, reviewing the original question and focusing solely on the ratio calculations:
Upon further investigation, since the numbers given do not accurately form a consistent geometric sequence (with a common ratio), the answer options you've provided may require a reevaluation in context. However, the calculated ratios lead to the conclusion that there is no valid response from the options given based on traditional geometric sequence definitions for this set of terms.
The correct determination here indicates that the sequence does not qualify under consistent geometric properties due to varying calculated ratios. Thus, if forced to pick, none of the suggested answers align with evident characteristics of consistent common ratios.
If you wish to focus solely on the ratio \( r = 6\) as an observed calculation from \( \frac{12}{2} \), please retain that metric as it is generally the ratio used for entries one to two but does not reflect continuous progression among other provided terms.
I apologize for the confusion. Let’s determine the common ratio between terms in the sequence \(2, 12, 18, 132, ...\) correctly and align it with one of your provided options.
For a geometric sequence, the common ratio \(r\) is found by dividing a term by the preceding term.
Let's calculate it:
From the first term to the second term: \[ r = \frac{12}{2} = 6 \]
From the second term to the third term: \[ r = \frac{18}{12} = 1.5 \]
From the third term to the fourth term: \[ r = \frac{132}{18} = \frac{132}{18} = 7.33 \quad (which is also not a consistent ratio) \]
It looks like this is not a geometric sequence since the ratios are not consistent. However, I realized that possibly there was a misunderstanding on the sequence definition given your choices.
Evaluating the choices again:
As no common ratio exists for this sequence as per standard geometric definitions, I would initially argue against a fitting response. Nonetheless, I can revisit the choice:
To find a feasible answer from the options provided related to progression, we note: Only \(4\) remains as an option we reframed based on indicative deviation:
Given that consecutive terms do not hold consistent ratios, if I must attribute to an option, I'd reference:
Thus emphasizing, the most logical answer given structure aligning to the options would be \(4\), but please recognize under longitudinal geometric scrutiny this remains dubious as consistently accurate.
To determine \(a_7\) for the sequence \(19, 15, 11, 7, 3, \ldots\), we first need to identify the pattern in the sequence.
The sequence starts with \(19\) and decreases consistently. We can find the common difference:
Thus, the common difference is \(-4\).
Now, we can write the general formula for the \(n\)-th term of an arithmetic sequence:
\[ a_n = a_1 + (n-1) \cdot d \] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In our case:
Now we can find \(a_7\):
\[ a_7 = 19 + (7-1) \cdot (-4) \] \[ a_7 = 19 + 6 \cdot (-4) \] \[ a_7 = 19 - 24 \] \[ a_7 = -5 \]
So, the answer is:
\(a_7 = -5\)
or using the subscript notation:
\(a_7 = -5\).
Given the recursively defined formula \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we want to determine the correct function notation.
To represent this in function notation, we can let \( f(n) \) represent the same sequence described by \( a_n \).
Initial Condition: Since \( a_1 = -3 \),
Recursive Formula: The recursive part \( a_n = a_{n-1} + 4 \) can be translated into function notation as:
Now, we can combine these two parts:
Therefore, the correct option is:
f(1) = -3 ; f(n) = f(n-1) + 4.
