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 , 10 .
Explanation
Identifying Arithmetic Sequences We need to identify which of the given sequences are arithmetic. An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. We will examine each sequence to determine if it meets this criterion.
Checking Each Sequence
3 , 7 , 11 , 15 , 19 : The differences between consecutive terms are 7 − 3 = 4 , 11 − 7 = 4 , 15 − 11 = 4 , and 19 − 15 = 4 . Since the difference is constant, this is an arithmetic sequence.
4 , 16 , 64 , 256 : The differences between consecutive terms are 16 − 4 = 12 , 64 − 16 = 48 , and 256 − 64 = 192 . Since the difference is not constant, this is not an arithmetic sequence.
48 , 24 , 12 , 6 , 3 , d o t s : The differences between consecutive terms are 24 − 48 = − 24 , 12 − 24 = − 12 , 6 − 12 = − 6 , and 3 − 6 = − 3 . Since the difference is not constant, this is not an arithmetic sequence.
1 , 4 , 9 , 16 , 25 , 36 : The differences between consecutive terms are 4 − 1 = 3 , 9 − 4 = 5 , 16 − 9 = 7 , 25 − 16 = 9 , and 36 − 25 = 11 . Since the difference is not constant, this is not an arithmetic sequence.
1 , 2 1 , 0 , − 2 1 : The differences between consecutive terms are 2 1 − 1 = − 2 1 , 0 − 2 1 = − 2 1 , and − 2 1 − 0 = − 2 1 . Since the difference is constant, this is an arithmetic sequence.
− 2 , 4 , − 8 , 16 , d o t s : The differences between consecutive terms are 4 − ( − 2 ) = 6 , − 8 − 4 = − 12 , and 16 − ( − 8 ) = 24 . Since the difference is not constant, this is not an arithmetic sequence.
1 , 0 , − 1 , − 2 , − 3 : The differences between consecutive terms are 0 − 1 = − 1 , − 1 − 0 = − 1 , − 2 − ( − 1 ) = − 1 , and − 3 − ( − 2 ) = − 1 . Since the difference is constant, this is an arithmetic sequence.
2 1 , 3 1 , 4 1 , 5 1 , d o t s : The differences between consecutive terms are 3 1 − 2 1 = − 6 1 , 4 1 − 3 1 = − 12 1 , and 5 1 − 4 1 = − 20 1 . Since the difference is not constant, this is not an arithmetic sequence.
3 x , x , 3 x , 9 x , d o t s : The differences between consecutive terms are x − 3 x = − 2 x , 3 x − x = − 3 2 x , and 9 x − 3 x = − 9 2 x . Since the difference is not constant, this is not an arithmetic sequence.
9.5 , 7.5 , 5.5 : The differences between consecutive terms are 7.5 − 9.5 = − 2 and 5.5 − 7.5 = − 2 . Since the difference is constant, this is an arithmetic sequence.
Identifying the Arithmetic Sequences Based on the analysis above, the arithmetic sequences are sequences 1, 5, 7, and 10.
Final Answer The arithmetic sequences are 1, 5, 7, and 10.
Examples
Arithmetic sequences are useful in various real-life scenarios, such as calculating simple interest, predicting patterns, and designing evenly spaced structures. For instance, if you deposit $100 into a savings account that earns $5 in simple interest each year, the amounts you have each year form an arithmetic sequence: $105, $110, $115, and so on. Understanding arithmetic sequences helps you predict how your savings will grow over time. Another example is in construction, where evenly spaced supports or beams follow an arithmetic sequence to ensure structural integrity.
The arithmetic sequences from the given options are sequences 1, 5, 7, and 10, as they all maintain a constant difference between consecutive terms. Sequences 2, 3, 4, 6, 8, and 9 do not have constant differences and therefore are not arithmetic sequences.
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