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Year 12 Maths

Arithmetic Series

Master the sum formula for arithmetic series, apply sigma notation, and solve real-world problems involving arithmetic sequences in HSC Advanced Mathematics.

The Sum Formula for Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. If the first term is a, the common difference is d, and the last term is l, the sum of n terms is given by:

Sn = n/2(a + l)

Where a is the first term, l is the last term, and n is the number of terms.

When the last term is not known, we can substitute l = a + (n − 1)d to get the alternative formula:

Sn = n/2(2a + (n − 1)d)

Example: Find the sum of the first 20 terms of the arithmetic series 3 + 7 + 11 + 15 + ...

Here a = 3, d = 4, n = 20.

S20 = 20/2(2(3) + (20 − 1)(4)) = 10(6 + 76) = 10 × 82 = 820

Sigma Notation

Sigma notation provides a compact way to express the sum of a series. The Greek letter Σ (sigma) means "sum of". We write:

Σk=1n (a + (k − 1)d) = Sn

The variable k is the index of summation, running from the lower to upper limits.

For instance, Σk=15 (2k + 1) means substitute k = 1, 2, 3, 4, 5 and add: 3 + 5 + 7 + 9 + 11 = 35.

Key properties of sigma notation:

Σk=1n c = cn (sum of a constant)

Σk=1n k = n(n+1)/2 (sum of first n natural numbers)

Σk=1n (ak + b) = aΣk + nb (linearity)

Applications of Arithmetic Series

Arithmetic series appear in many real-world contexts, including financial problems, pattern counting, and physical phenomena.

Salary growth: A graduate starts on $50,000 and receives a $3,000 raise each year. What is their total earnings over 10 years?

a = 50000, d = 3000, n = 10. S10 = 10/2(2 × 50000 + 9 × 3000) = 5(100000 + 27000) = $635,000

Another common application involves counting objects arranged in patterns, such as seats in a stadium where each row has more seats than the previous.

Stadium seating: The first row has 20 seats, and each subsequent row has 2 more seats. How many seats are there in the first 30 rows?

a = 20, d = 2, n = 30. S30 = 30/2(2 × 20 + 29 × 2) = 15(40 + 58) = 15 × 98 = 1,470 seats

Key Vocabulary

Arithmetic Series

The sum of the terms in an arithmetic sequence, where each term differs from the previous by a constant common difference.

Common Difference

The constant value d added to each term to produce the next term in an arithmetic sequence.

Sigma Notation

A compact notation using the symbol Σ to represent the sum of a series, with upper and lower limits of summation.

Partial Sum

The sum of the first n terms of a series, denoted Sn.

Worked Examples

1

Find the sum of the arithmetic series 5 + 8 + 11 + ... + 101.

Step 1: Identify a = 5, d = 3, l = 101.

Step 2: Find n: l = a + (n − 1)d, so 101 = 5 + (n − 1)(3), giving 96 = 3(n − 1), so n = 33.

Answer: S33 = 33/2(5 + 101) = 33/2 × 106 = 1,749

2

Evaluate Σk=150 (4k − 1).

Step 1: This is an arithmetic series with first term (k=1): 4(1) − 1 = 3, and last term (k=50): 4(50) − 1 = 199.

Step 2: n = 50, a = 3, l = 199.

Answer: S50 = 50/2(3 + 199) = 25 × 202 = 5,050

3

An arithmetic series has S10 = 275 and a = 5. Find the common difference d.

Step 1: Use Sn = n/2(2a + (n − 1)d): 275 = 10/2(2(5) + 9d).

Step 2: 275 = 5(10 + 9d), so 55 = 10 + 9d, giving 9d = 45.

Answer: d = 5

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Find the sum of the first 15 terms of the arithmetic series where a = 2 and d = 3.

Question 2

How many terms are in the arithmetic series 7 + 11 + 15 + ... + 203?

Question 3

Evaluate Σk=120 (3k + 2).

Question 4

An arithmetic series has first term 10 and common difference −2. What is S8?

Question 5

The sum of the first n terms of an arithmetic series is given by Sn = 3n2 + 2n. What is the 10th term?

Key Concepts Summary

Integration Geometric Series