Arithmetic and Geometric Sequences

Arithmetic and geometric sequences are two fundamental types of numerical progressions. They describe how quantities grow or shrink by addition or by multiplication, and they form the foundation for topics such as series, summation formulas, and exponential growth.

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1. Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount called the common difference d.

a, a + d, a + 2d, a + 3d, … , a + (n − 1)d

• a – first term
• d – common difference

The nth term, denoted T_n, is given by:

T_n = a + (n − 1)d

Each new term is obtained by adding d to the previous term. The difference between consecutive terms remains constant:

T_k+1 − T_k = d

Example: If a = 4 and d = 3, the sequence is 4, 7, 10, 13, 16, … The 20th term is T₂₀ = 4 + (20 − 1)×3 = 61.

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2. Geometric Sequence

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant called the common ratio r.

a, a·r, a·r², a·r³, … , a·rⁿ⁻¹

• a – first term
• r – common ratio

The nth term is given by:

T_n = a·rⁿ⁻¹

The ratio between any two consecutive terms remains constant:

T_k+1 / T_k = r

Example: If a = 5 and r = 2, the sequence is 5, 10, 20, 40, 80, … The 8th term is T₈ = 5·2⁷ = 640.

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3. Comparison

• In an arithmetic sequence, the change is additive — the same amount is added or subtracted each time.
• In a geometric sequence, the change is multiplicative — each term is multiplied by the same factor.
• Both follow precise patterns that allow direct computation of any term without listing the entire sequence.

Arithmetic: T_n = a + (n − 1)d
Geometric: T_n = a·rⁿ⁻¹

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4. From Sequences to Series

When the terms of a sequence are added together, they form a series. The formulas for arithmetic and geometric series are direct consequences of the patterns above and are essential tools in algebra and calculus for analysing sums, limits, and convergence.

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