Which of the following sequences is an arithmetic sequence and why?
1. $3,7,11,15,19$
2. $4,16,64,256$
3. $48,24,12,6,3, \ldots$
4. $1,4,9,16,25,36$
5. $1, \frac{1}{2}, 0,-\frac{1}{2}$
6. $-2,4,-8,16, \ldots$
7. $1,0,-1,-2,-3$
8. $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots$
9. $3 x, x, \frac{x}{3}, \frac{x}{9}, \ldots$
10. $9.5, 7.5, 5.5$
Answer (2)
An arithmetic sequence has a constant difference between consecutive terms.
Sequence 1: 3 , 7 , 11 , 15 , 19 has a common difference of 4.
Sequence 5: 1 , 2 1 , 0 , − 2 1 has a common difference of − 2 1 .
Sequence 7: 1 , 0 , − 1 , − 2 , − 3 has a common difference of -1.
Sequence 10: 9.5 , 7.5 , 5.5 has a common difference of -2.
The arithmetic sequences are 1, 5, 7, and 10, so the answer is 1 , 5 , 7 , 10 .
Explanation
Understanding Arithmetic Sequences We need to determine which of the given sequences are arithmetic sequences. An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
Identifying Arithmetic Sequences
3 , 7 , 11 , 15 , 19 : The difference between consecutive terms is 7 − 3 = 4 , 11 − 7 = 4 , 15 − 11 = 4 , 19 − 15 = 4 . The difference is constant, so this is an arithmetic sequence.
4 , 16 , 64 , 256 : The difference between consecutive terms is 16 − 4 = 12 , 64 − 16 = 48 . The difference is not constant, so this is not an arithmetic sequence.
48 , 24 , 12 , 6 , 3 , … : The difference between consecutive terms is 24 − 48 = − 24 , 12 − 24 = − 12 . The difference is not constant, so this is not an arithmetic sequence.
1 , 4 , 9 , 16 , 25 , 36 : The difference between consecutive terms is 4 − 1 = 3 , 9 − 4 = 5 . The difference is not constant, so this is not an arithmetic sequence.
1 , 2 1 , 0 , − 2 1 : The difference between consecutive terms is 2 1 − 1 = − 2 1 , 0 − 2 1 = − 2 1 , − 2 1 − 0 = − 2 1 . The difference is constant, so this is an arithmetic sequence.
− 2 , 4 , − 8 , 16 , … : The difference between consecutive terms is 4 − ( − 2 ) = 6 , − 8 − 4 = − 12 . The difference is not constant, so this is not an arithmetic sequence.
1 , 0 , − 1 , − 2 , − 3 : The difference between consecutive terms is 0 − 1 = − 1 , − 1 − 0 = − 1 , − 2 − ( − 1 ) = − 1 , − 3 − ( − 2 ) = − 1 . The difference is constant, so this is an arithmetic sequence.
2 1 , 3 1 , 4 1 , 5 1 , … : The difference between consecutive terms is 3 1 − 2 1 = − 6 1 , 4 1 − 3 1 = − 12 1 . The difference is not constant, so this is not an arithmetic sequence.
3 x , x , 3 x , 9 x , … : The difference between consecutive terms is x − 3 x = − 2 x , 3 x − x = − 3 2 x . The difference is not constant, so this is not an arithmetic sequence.
9.5 , 7.5 , 5.5 : The difference between consecutive terms is 7.5 − 9.5 = − 2 , 5.5 − 7.5 = − 2 . The difference is constant, so this is an arithmetic sequence.
Final Answer The arithmetic sequences are sequences 1, 5, 7, and 10.
Examples
Arithmetic sequences are useful in various real-life scenarios, such as calculating simple interest, predicting evenly spaced events, or designing patterns with a constant increment. For instance, if you save a fixed amount of money each month, the total savings over time form an arithmetic sequence. Understanding arithmetic sequences helps in making predictions and managing resources effectively in these situations.
The arithmetic sequences from the given options are 1, 5, 7, and 10. Each of these sequences has a constant difference between consecutive terms. Sequences 2, 3, 4, 6, 8, and 9 do not have a constant difference and are not arithmetic sequences.
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