Responses
Add the common difference of −3.5
Multiply by the common ratio of −3.5
Multiply by the common difference of −3.5
Add the common ratio of −3.5
9 answers
To determine the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, we can first find the difference between consecutive terms:
We see that the difference is consistently \(-3.5\).
Therefore, we can conclude that the rule for generating the next term is:
Add the common difference of −3.5.
So, the correct response is: Add the common difference of −3.5.
To classify the given list of numbers \(9, 4, -1\), we can check if it forms an arithmetic sequence or a geometric sequence.
Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.
Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.
Since the sequence has a constant difference, we classify the list of numbers as an arithmetic sequence.
So the correct response is: arithmetic sequence.
To find the common ratio in a geometric sequence, we can divide any term by the preceding term. Let's calculate the common ratio using the first two terms of the given sequence \(2, \frac{1}{2}\):
\[ \text{Common Ratio (r)} = \frac{\text{Second Term}}{\text{First Term}} = \frac{\frac{1}{2}}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
Now, let's check this ratio with the next set of terms for confirmation:
From \(\frac{1}{2}\) to \(\frac{1}{8}\): \[ \frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{8} \times \frac{2}{1} = \frac{2}{8} = \frac{1}{4} \]
From \(\frac{1}{8}\) to \(\frac{1}{32}\): \[ \frac{\frac{1}{32}}{\frac{1}{8}} = \frac{1}{32} \times \frac{8}{1} = \frac{8}{32} = \frac{1}{4} \]
In each case, the common ratio \(r\) equals \(\frac{1}{4}\).
Therefore, the common ratio of the sequence is \(\frac{1}{4}\).
So the correct response is: 1/4.
To find \( a_7 \) in the sequence \( 19, 15, 11, 7, 3 \), we first need to determine the rule governing the sequence.
The sequence is decreasing, and we can calculate the difference between consecutive terms:
The common difference is \( -4 \) (we subtract \( 4 \) each time).
Using the formula for the \( n \)-th term of an arithmetic sequence: \[ a_n = a_1 + (n - 1) \cdot d \] where:
Now, we can substitute these values into the formula to find \( a_7 \): \[ a_7 = 19 + (7 - 1) \cdot (-4) \] \[ a_7 = 19 + 6 \cdot (-4) \] \[ a_7 = 19 - 24 \] \[ a_7 = -5 \]
Thus, the value for \( a_7 \) is \(-5\).
The correct response is: a7=−5.
To interpret the recursively defined formula given as \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we need to express this correctly in function notation.
Let's analyze the responses:
\( f(1) = -3; f(n) = a_{n-1} + 4 \)
This is incorrect because \( f(n) \) should reference \( f(n-1) \), not \( a_{n-1} \).
\( f(1) = -3; f(n) = f(n-1) + 4 \)
This is correct. It represents the initial condition and the recursive definition correctly in terms of the function \( f \).
\( a(1) = -3; f(n) = f(n-1) + 4 \)
This is incorrect because it mixes \( a \) notation with \( f \) notation.
\( f(1) = -3; f(n) = f(a_{n-1}) + 4 \)
This is incorrect because it incorrectly refers to \( a_{n-1} \) instead of \( f(n-1) \).
Based on this analysis, the correct option that represents the recursive definition in function notation is:
\( f(1) = -3; f(n) = f(n-1) + 4 \).
