If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 +

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)

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)

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 5: Expand S_(n-1): S_(n-1) = 3(n^2 - 2n + 1) + 2(n - 1).
Step 6: Simplify S_(n-1): S_(n-1) = 3n^2 - 6n + 3 + 2n - 2 = 3n^2 - 4n + 1.
Step 7: Now, find the difference S_n - S_(n-1): (3n^2 + 2n) - (3n^2 - 4n + 1).
Step 8: Simplify the difference: 3n^2 + 2n - 3n^2 + 4n - 1 = 6n - 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.

Arithmetic Series – Understanding the properties of arithmetic series, including the formula for the sum of the first n terms.
Difference of Sums – Calculating the common difference by finding the difference between consecutive sums of the series.
Quadratic Functions – Recognizing and manipulating quadratic expressions to derive necessary values.