Use the arithmetic series [tex]$14+6+(-2)+(-10)+(-18)+(-26)+(-34)$[/tex] to answer each question. What is the common difference [tex]$d$[/tex] of the arithmetic sequence on which the series is based? [tex]$d=\square$[/tex]
Answer (2)
The common difference d is found by subtracting a term from its subsequent term.
Calculate d = 6 − 14 = − 8 .
Verify the common difference with another pair of consecutive terms: d = − 2 − 6 = − 8 .
The common difference of the arithmetic sequence is − 8 .
Explanation
Understanding the Problem We are given the arithmetic series 14 + 6 + ( − 2 ) + ( − 10 ) + ( − 18 ) + ( − 26 ) + ( − 34 ) . Our goal is to find the common difference d of the arithmetic sequence on which this series is based. The common difference is the constant value that is added to each term to get the next term in the sequence.
Calculating the Common Difference To find the common difference d , we can subtract any term from its subsequent term. Let's subtract the first term from the second term: d = 6 − 14 = − 8 We can verify this by subtracting the second term from the third term: d = − 2 − 6 = − 8 And also by subtracting the third term from the fourth term: d = − 10 − ( − 2 ) = − 10 + 2 = − 8
Final Answer Thus, the common difference d of the arithmetic sequence is − 8 .
Examples
Arithmetic sequences and series are useful in various real-life scenarios. For example, consider a savings plan where you deposit a fixed amount of money each month. If you deposit $100 in the first month, $150 in the second month, $200 in the third month, and so on, this forms an arithmetic sequence. Understanding the common difference (in this case, $50) helps you predict your savings over time. Similarly, in calculating depreciation of an asset or determining the number of seats in rows of a stadium, arithmetic sequences can be very helpful.
The common difference d of the arithmetic sequence is − 8 , found by subtracting each term from the next. This was verified using multiple pairs of consecutive terms. Thus, d = − 8 .
;
