What is the relationship between the successive terms in the sequence shown?
[tex]$9,-1,-11,-21, \ldots$[/tex]
A. The common difference is -10.
B. The common difference is 10.
C. The common ratio is -9.
D. The common ratio is 9.
Answer (2)
Calculate the difference between consecutive terms: − 1 − 9 = − 10 , − 11 − ( − 1 ) = − 10 , − 21 − ( − 11 ) = − 10 .
Calculate the ratio between consecutive terms: 9 − 1 , − 1 − 11 , − 11 − 21 .
Since the difference between consecutive terms is constant (-10), the sequence has a common difference.
The common difference is -10, so the answer is: $\boxed{\text{The common difference is -10 .}}
Explanation
Analyzing the Problem We are given the sequence 9 , − 1 , − 11 , − 21 , … and asked to determine the relationship between successive terms. The options are:
The common difference is -10.
The common difference is 10.
The common ratio is -9.
The common ratio is 9.
Calculating the Common Difference To find the relationship, we will calculate the difference and ratio between consecutive terms.
Difference between the first two terms: − 1 − 9 = − 10 Difference between the second and third terms: − 11 − ( − 1 ) = − 11 + 1 = − 10 Difference between the third and fourth terms: − 21 − ( − 11 ) = − 21 + 11 = − 10
Since the difference between consecutive terms is constant, the sequence has a common difference. The common difference is -10.
Calculating the Common Ratio Ratio between the first two terms: 9 − 1 = − 9 1 Ratio between the second and third terms: − 1 − 11 = 11 Ratio between the third and fourth terms: − 11 − 21 = 11 21
Since the ratio between consecutive terms is not constant, the sequence does not have a common ratio.
Determining the Relationship The common difference is -10, which matches the first option.
Examples
Consider a scenario where the temperature decreases by 10 degrees every hour. If the initial temperature is 9 degrees, the sequence of temperatures would be 9, -1, -11, -21, and so on. This is an arithmetic sequence with a common difference of -10. Understanding arithmetic sequences helps in predicting future temperatures or analyzing trends in temperature changes.
The sequence 9 , − 1 , − 11 , − 21 , … has a common difference of -10 between successive terms. There is no common ratio as the ratios differ between terms. Therefore, the correct choice is A.
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