Mathematics

Mathematics

Problem

1. Find the sum of the series 2+25+225+…+2542+\frac{2}{5}+\frac{2}{25}+\ldots+\frac{2}{5^{4}}.
Intuitive explanation

This is a finite geometric series where each term is a constant multiple of the previous one. Notice that after the first term, each new term equals the previous term multiplied by 15\frac{1}{5} — that's your common ratio. The key insight is to use the geometric series formula, which lets you avoid adding five terms individually and instead find the sum in one clean calculation.

Intuitive explanation

This is a finite geometric series where each term is a constant multiple of the previous one. Notice that after the first term, each new term equals the previous term multiplied by 15\frac{1}{5} — that's your common ratio. The key insight is to use the geometric series formula, which lets you avoid adding five terms individually and instead find the sum in one clean calculation.

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