If the sum of the first n terms of an arithmetic progression is given by S_n = 3

If the sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 + 2n, what is the common difference?

If the sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 + 2n, what is the common difference?

Step 1: Understand that S_n represents the sum of the first n terms of an arithmetic progression (AP).
Step 2: The formula for the sum of the first n terms of an AP is S_n = n/2 * (2a + (n-1)d), where a is the first term and d is the common difference.
Step 3: We are given S_n = 3n^2 + 2n. We need to find the common difference d.
Step 4: To find the common difference, we can differentiate S_n with respect to n. This gives us the formula for the nth term of the AP.
Step 5: Differentiate S_n = 3n^2 + 2n. The derivative is dS_n/dn = 6n + 2.
Step 6: The nth term of the AP, T_n, can be found using T_n = S_n - S_(n-1).
Step 7: Calculate S_(n-1) by substituting (n-1) into the original S_n formula: S_(n-1) = 3(n-1)^2 + 2(n-1).
Step 8: Simplify S_(n-1) to find its value.
Step 9: Now, find T_n = S_n - S_(n-1). This will give you the nth term of the AP.
Step 10: The common difference d is the difference between consecutive terms, which can be found from T_n and T_(n-1).

Arithmetic Progression (AP) – An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant.
Sum of n Terms – The sum of the first n terms of an arithmetic progression can be expressed using the formula S_n = n/2 * (2a + (n-1)d), where a is the first term and d is the common difference.
Differentiation – Differentiating the sum function S_n with respect to n helps in finding the common difference of the arithmetic progression.