If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the common difference? (2023)
Practice Questions
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If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the common difference? (2023)
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The common difference can be found by calculating S_n - S_(n-1). Here, S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). The difference simplifies to 4.
Questions & Step-by-step Solutions
1 item
Q
Q: If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the common difference? (2023)
Solution: The common difference can be found by calculating S_n - S_(n-1). Here, S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). The difference simplifies to 4.
Steps: 10
Step 1: Understand that S_n represents the sum of the first n terms of the arithmetic series.
Step 2: Write down the formula for S_n, which is given as S_n = 3n^2 + 2n.
Step 3: To find the common difference, we need to calculate S_(n-1), which is the sum of the first (n-1) terms.
Step 4: Substitute (n-1) into the formula: S_(n-1) = 3(n-1)^2 + 2(n-1).
Step 9: The common difference is the result of S_n - S_(n-1), which is 6n - 1.
Step 10: Since the common difference is constant in an arithmetic series, we can find it by evaluating at a specific n, for example, n=1: 6(1) - 1 = 5.
