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 10th term of the series? (2021)

If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the 10th term of the series? (2021)

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 = 3n^2 + 2n.
Step 3: To find the 10th term (T_10), we need to calculate S_10 and S_9.
Step 4: Calculate S_10 by substituting n = 10 into the formula: S_10 = 3(10^2) + 2(10).
Step 5: Calculate 10^2, which is 100. Then, multiply by 3: 3 * 100 = 300.
Step 6: Calculate 2 * 10, which is 20.
Step 7: Add the results from Step 5 and Step 6: S_10 = 300 + 20 = 320.
Step 8: Now calculate S_9 by substituting n = 9 into the formula: S_9 = 3(9^2) + 2(9).
Step 9: Calculate 9^2, which is 81. Then, multiply by 3: 3 * 81 = 243.
Step 10: Calculate 2 * 9, which is 18.
Step 11: Add the results from Step 9 and Step 10: S_9 = 243 + 18 = 261.
Step 12: Now, find the 10th term using the formula T_n = S_n - S_(n-1): T_10 = S_10 - S_9.
Step 13: Substitute the values: T_10 = 320 - 261.
Step 14: Calculate the result: T_10 = 59.

Arithmetic Series and Sequences – Understanding the properties of arithmetic series, including how to derive the nth term from the sum of the first n terms.
Summation Formulas – Applying the formula for the sum of an arithmetic series to find specific terms.
Difference of Sums – Using the relationship between consecutive sums to find individual terms in a series.