If the sum of the first n terms of an arithmetic series is given by S_n = 5n + 3

If the sum of the first n terms of an arithmetic series is given by S_n = 5n + 3, what is the common difference? (2023)

If the sum of the first n terms of an arithmetic series is given by S_n = 5n + 3, 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: The formula for S_n is given as S_n = 5n + 3.
Step 3: To find the common difference, we need to calculate S_n - S_(n-1).
Step 4: First, calculate S_(n-1) by substituting (n-1) into the formula: S_(n-1) = 5(n-1) + 3.
Step 5: Simplify S_(n-1): S_(n-1) = 5n - 5 + 3 = 5n - 2.
Step 6: Now, calculate S_n - S_(n-1): S_n - S_(n-1) = (5n + 3) - (5n - 2).
Step 7: Simplify the expression: S_n - S_(n-1) = 5n + 3 - 5n + 2 = 5.
Step 8: The result, 5, is the common difference of the arithmetic series.

Arithmetic Series – An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant.
Sum of Terms – The sum of the first n terms of an arithmetic series can be expressed in a formula that relates to the first term and the common difference.
Finding Common Difference – The common difference can be derived from the sum of terms by calculating the difference between consecutive sums.