Arithmetic Sequences

An arithmetic sequence is a list of numbers where consecutive terms differ by a constant amount called the common difference (d).

General Term (n-th term)

T_n = a + (n − 1)d

Sum of First n Terms

S_n = n/2 × (2a + (n−1)d)

a = first term, d = common difference, n = term number

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Worked Example: Arithmetic sequence

For the sequence 3, 7, 11, 15, …, find the 20th term and the sum of the first 20 terms.

a = 3, d = 7 − 3 = 4, n = 20

20th term: T₂₀ = 3 + (20 − 1) × 4 = 3 + 76 = 79

Sum: S₂₀ = 20/2 × (2 × 3 + 19 × 4) = 10 × (6 + 76) = 10 × 82 = 820

Arithmetic sequence: 3, 7, 11, 15, 19, …

3 +4 7 +4 11 +4 15 +4 19 …

Geometric Sequences

A geometric sequence is a list of numbers where each term is multiplied by a constant called the common ratio (r).

General Term

T_n = a × rⁿ⁻¹

Sum of First n Terms (r ≠ 1)

S_n = a(rⁿ − 1) / (r − 1)

a = first term, r = common ratio, n = term number

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Worked Example: Geometric sequence

For the sequence 2, 6, 18, 54, …, find the 8th term and the sum of the first 6 terms.

a = 2, r = 6/2 = 3, n = 8

8th term: T₈ = 2 × 3⁷ = 2 × 2187 = 4374

Sum of 6 terms: S₆ = 2(3⁶ − 1) / (3 − 1) = 2(729 − 1) / 2 = 728 = 728

Geometric sequence: 2, 6, 18, 54, 162, …

2 ×3 6 ×3 18 ×3 54 ×3 162 …

Identifying and Applying Sequences

Test for Arithmetic

Subtract consecutive terms: T₂ − T₁ = T₃ − T₂ = d (constant)

Test for Geometric

Divide consecutive terms: T₂ / T₁ = T₃ / T₂ = r (constant)

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Worked Example: Real-world application

A savings plan deposits $100 in month 1, $150 in month 2, $200 in month 3, and so on. How much is deposited in month 12? What is the total deposited after 12 months?

This is arithmetic: a = 100, d = 50, n = 12

Month 12: T₁₂ = 100 + 11 × 50 = 100 + 550 = $650

Total (S₁₂): = 12/2 × (2 × 100 + 11 × 50) = 6 × (200 + 550) = 6 × 750 = $4,500

Key Vocabulary

Common Difference (d)

The fixed amount added between terms in an arithmetic sequence. Can be positive (increasing) or negative (decreasing).

Common Ratio (r)

The fixed multiplier between terms in a geometric sequence. r > 1 gives growth; 0 < r < 1 gives decay.

Series

The sum of the terms of a sequence. The sum of the first n terms is called the n-th partial sum, written S_n.

Sigma Notation (Σ)

A compact way to write a series. Σ means “sum of”. For example, Σ (from k=1 to n) of T_k = S_n.

Knowledge Check

Test your knowledge of arithmetic and geometric sequences and series.

Question 1

For the arithmetic sequence 5, 9, 13, 17, …, what is the common difference?

3 5 4 9

Question 2

For the geometric sequence 4, 12, 36, 108, …, what is the common ratio?

4 8 2 3

Question 3

Find the 10th term of the arithmetic sequence with a = 2 and d = 6.

56 60 62 54

Question 4

Find the sum of the first 5 terms of the geometric sequence: 1, 2, 4, 8, 16, …

16 31 32 63

Question 5

An arithmetic series has a = 5, d = 3, and S_n = 155. How many terms are in the series?

8 12 10 15

Sequences & Series