Unit 3: The Sum of Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. If the sequence is:

a, (a + d), (a + 2d), ..., (a + (n-1)d)

Then the series is:

S_n = a + (a + d) + (a + 2d) + ... + l

General Formula

The sum of the first n terms of an arithmetic series is given by:

S_n = (n/2)[2a + (n - 1)d]

Where:

• a is the first term.
• d is the common difference.
• n is the number of terms.

Alternative Formula

If the last term l is known:

S_n = (n/2)(a + l)

Where:

• l is the last term, calculated as l = a + (n - 1)d.

1. Identify the values of a, d, and n from the sequence.
2. Select the appropriate formula based on the known values.
3. Substitute the values into the formula and solve for the sum.

Example 1

Problem: Find the sum of the first 10 terms of the arithmetic series 3, 7, 11, ...

Solution:

• First term, a = 3
• Common difference, d = 7 - 3 = 4
• Number of terms, n = 10

Using the formula:

S_n = (n/2)[2a + (n - 1)d]

Substituting values:

S₁₀ = (10/2)[2 × 3 + (10 - 1) × 4] = 5[6 + 36] = 5 × 42 = 210

Therefore, the sum of the first 10 terms is 210.

Example 2

Problem: Given an arithmetic series with a = 4, d = 7, and l = 368, find S_n.

Solution:

First, find n:

l = a + (n - 1)d ⇒ 368 = 4 + (n - 1) × 7

Solve for n:

n - 1 = (368 - 4)/7 ⇒ n - 1 = 52 ⇒ n = 53

Using the alternative formula:

S_n = (n/2)(a + l)

Substituting values:

S₅₃ = (53/2)(4 + 368) = (53/2)(372) = 53 × 186 = 9,858

Therefore, the sum of the series is 9,858.

1. Find the sum of the first 15 terms of the arithmetic series 2, 5, 8, ...
2. If the first term is 7, the common difference is 3, and the last term is 64, find the sum of the series.
3. Determine the sum of all even numbers between 2 and 100.
4. The 5th term of an arithmetic series is 20, and the 10th term is 35. Find the sum of the first 10 terms.

1. What is the sum of the first 5 terms of the arithmetic series 1, 3, 5, 7, ...?

15

25

35

45

2. If a = 4, d = 6, and n = 10, what is S_n?

100

280

320

400

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• An arithmetic series is the sum of the terms in an arithmetic sequence.
• The sum can be found using formulas based on known values.
• Understanding arithmetic series is important for advanced mathematical concepts and real-world applications.