Determine if the given sequences are arithmetic sequences or not. If yes, then what is the difference?
1. $-11, -7, -3, 1 \ldots$
A. No
B. Yes, $d = 4$
C. Yes, $d = -4$
D. Yes, $d = 5$
2. $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \ldots$
A. No
B. Yes, $d = \frac{1}{3}$
C. Yes, $d = \frac{1}{6}$
D. Yes, $d = 1$
Answer (1)
Sequence 2.012 has a common difference of 4, so it is arithmetic.
Sequence 2.013 does not have a common difference, so it is not arithmetic.
The answer for sequence 2.012 is Yes, d = 4 .
The answer for sequence 2.013 is No.
Explanation
Understanding Arithmetic Sequences We are given two sequences and asked to determine if they are arithmetic sequences. If a sequence is arithmetic, we need to find the common difference, denoted by 'd'. An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
Analyzing Sequence 2.012 For sequence 2.012: -11, -7, -3, 1, ... We calculate the difference between consecutive terms: -7 - (-11) = 4 -3 - (-7) = 4 1 - (-3) = 4 Since the difference between consecutive terms is constant and equal to 4, the sequence is arithmetic with a common difference of 4.
Analyzing Sequence 2.013 For sequence 2.013: 2 1 , 3 1 , 4 1 , 5 1 , ... We calculate the difference between consecutive terms: 3 1 − 2 1 = 6 2 − 6 3 = − 6 1 4 1 − 3 1 = 12 3 − 12 4 = − 12 1 5 1 − 4 1 = 20 4 − 20 5 = − 20 1 Since the difference between consecutive terms is not constant, the sequence is not arithmetic.
Conclusion Therefore, sequence 2.012 is arithmetic with d = 4, and sequence 2.013 is not arithmetic.
Examples
Arithmetic sequences are useful in various real-life scenarios, such as calculating simple interest, predicting the number of seats in successive rows of a theater, or determining patterns in evenly spaced objects. For example, if you save $100 each month, the total amount saved over several months forms an arithmetic sequence. Understanding arithmetic sequences helps in making predictions and managing finances effectively.
