Exploring Series in Numbers and Sequences

What are Series?

A series is the sum of terms of a sequence. Here are the main types:

Arithmetic Series

The sum of terms in an Arithmetic Progression (AP).

Formula: S_n = (n/2) (2a + (n-1)d)

Example: 2 + 4 + 6 + 8 + 10

Geometric Series

The sum of terms in a Geometric Progression (GP).

Formula: S_n = a (1 - rⁿ) / (1 - r), r ≠ 1

Example: 2 + 4 + 8 + 16

Harmonic Series

The sum of reciprocals of natural numbers.

Example: 1 + 1/2 + 1/3 + 1/4 + ...

Interactive Section

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Applications of Series

Arithmetic Series

• Finance: Calculating loan payments or savings over time.
• Construction: Counting the number of tiles or bricks required.

Geometric Series

• Technology: Exponential growth of storage or data.
• Physics: Compound interest or radioactive decay.

Harmonic Series

• Music: Frequencies in sound waves.
• Mathematics: Series approximations and number theory.