What is the common difference for this arithmetic sequence? 31,48,65,82,…
The Correct answer and Explanation is:
To find the common difference of the arithmetic sequence given: 31, 48, 65, 82, …, we start by understanding what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is known as the common difference.
Step-by-Step Calculation:
- Identify the Terms: The first few terms of the sequence are:
- First term (a₁) = 31
- Second term (a₂) = 48
- Third term (a₃) = 65
- Fourth term (a₄) = 82
- Calculate the Differences: To find the common difference (d), we subtract the first term from the second term, the second from the third, and so forth.
- From a₁ to a₂:d=a2−a1=48−31=17d = a₂ – a₁ = 48 – 31 = 17d=a2−a1=48−31=17
- From a₂ to a₃:d=a3−a2=65−48=17d = a₃ – a₂ = 65 – 48 = 17d=a3−a2=65−48=17
- From a₃ to a₄:d=a4−a3=82−65=17d = a₄ – a₃ = 82 – 65 = 17d=a4−a3=82−65=17
- Confirm the Consistency: As calculated, the difference between each pair of consecutive terms is consistently 17. This indicates that the sequence is indeed arithmetic, as the common difference remains unchanged across the terms.
- Conclusion: Therefore, the common difference ddd for the arithmetic sequence 31, 48, 65, 82, … is 17.
Significance of the Common Difference:
The common difference plays a crucial role in defining the behavior of the arithmetic sequence. It determines how quickly the sequence grows or shrinks. A positive common difference, like in this case, means the sequence is increasing. Understanding this concept is vital in various mathematical applications, including series and sequences, and helps in predicting future terms of the sequence by simply adding the common difference repeatedly to the last term.
In summary, the common difference for the given arithmetic sequence is 17, and this fundamental aspect allows us to analyze and extend the sequence efficiently.