If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the 4th term?

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Q: If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the 4th term?

Solution: The 4th term a_4 = S_4 - S_3 = (3(4^2) + 2(4)) - (3(3^2) + 2(3)) = (48 + 8) - (27 + 6) = 56 - 33 = 23.

Steps: 10

Step 1: Understand that S_n represents the sum of the first n terms of the arithmetic series.
Step 2: To find the 4th term (a_4), we need to calculate S_4 (the sum of the first 4 terms) and S_3 (the sum of the first 3 terms).
Step 3: Calculate S_4 using the formula S_n = 3n^2 + 2n. Substitute n = 4: S_4 = 3(4^2) + 2(4).
Step 4: Calculate 4^2, which is 16. Then, multiply by 3: 3 * 16 = 48.
Step 5: Calculate 2(4), which is 8. Now add the two results: 48 + 8 = 56. So, S_4 = 56.
Step 6: Now calculate S_3 using the same formula. Substitute n = 3: S_3 = 3(3^2) + 2(3).
Step 7: Calculate 3^2, which is 9. Then, multiply by 3: 3 * 9 = 27.
Step 8: Calculate 2(3), which is 6. Now add the two results: 27 + 6 = 33. So, S_3 = 33.
Step 9: Now, find the 4th term a_4 by subtracting S_3 from S_4: a_4 = S_4 - S_3 = 56 - 33.
Step 10: Calculate 56 - 33, which equals 23. Therefore, the 4th term a_4 is 23.