If the sum of the first n terms of an arithmetic series is given by S_n = 5n^2 +

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Question: If the sum of the first n terms of an arithmetic series is given by S_n = 5n^2 + 3n, what is the 5th term?

Options:

Correct Answer: 38

Solution:

The 5th term can be found using a_n = S_n - S_(n-1). Calculate S_5 and S_4, then find a_5 = S_5 - S_4 = (5(5^2) + 3(5)) - (5(4^2) + 3(4)) = 38.

If the sum of the first n terms of an arithmetic series is given by S_n = 5n^2 + 3n, what is the 5th term?

If the sum of the first n terms of an arithmetic series is given by S_n = 5n^2 + 3n, what is the 5th 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 = 5n^2 + 3n.
Step 3: To find the 5th term (a_5), we need to calculate S_5 and S_4.
Step 4: Calculate S_5 by substituting n = 5 into the formula: S_5 = 5(5^2) + 3(5).
Step 5: Simplify S_5: S_5 = 5(25) + 15 = 125 + 15 = 140.
Step 6: Calculate S_4 by substituting n = 4 into the formula: S_4 = 5(4^2) + 3(4).
Step 7: Simplify S_4: S_4 = 5(16) + 12 = 80 + 12 = 92.
Step 8: Now, find the 5th term a_5 using the formula a_n = S_n - S_(n-1): a_5 = S_5 - S_4.
Step 9: Substitute the values: a_5 = 140 - 92.
Step 10: Simplify to find a_5: a_5 = 48.

Arithmetic Series – Understanding the formula for the sum of the first n terms and how to derive individual terms from it.
Difference of Sums – Using the relationship between the sum of terms to find individual terms in a sequence.