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

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Question: The sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n. What is the 10th term?

Options:

Correct Answer: 36

Solution:

The nth term a_n = S_n - S_(n-1) = (3n^2 + 2n) - (3(n-1)^2 + 2(n-1)). For n=10, a_10 = 32.

The sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n. What is the 10th term?

The sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n. What is the 10th term?

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 S_n = 3n^2 + 2n.
Step 3: To find the nth term (a_n), use the formula a_n = S_n - S_(n-1).
Step 4: Calculate S_(n-1) by substituting (n-1) into the S_n 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) = 3n^2 - 6n + 3 + 2n - 2 = 3n^2 - 4n + 1.
Step 6: Now substitute S_n and S_(n-1) into the formula for a_n: a_n = (3n^2 + 2n) - (3n^2 - 4n + 1).
Step 7: Simplify the expression: a_n = 3n^2 + 2n - 3n^2 + 4n - 1 = 6n - 1.
Step 8: To find the 10th term (a_10), substitute n = 10 into the formula: a_10 = 6(10) - 1.
Step 9: Calculate a_10: a_10 = 60 - 1 = 59.

Arithmetic Series – Understanding the properties and formulas related to the sum of terms in an arithmetic series.
Difference of Sums – Using the relationship between the sum of terms and the individual terms to find specific terms in a series.
Quadratic Functions – Recognizing and manipulating quadratic expressions to derive values.