Subjects algebra

Fraction Root Pattern

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1. **Stating the problem:** We are given a sequence of numbers arranged in a stepped horizontal layout involving integers, fractions, and square roots: starting with 2, then fractions like $\frac{1}{1}$ and $\frac{1}{2\sqrt{5}}$, arranged diagonally downwards. 2. **Understanding the elements:** The numbers include integers (2), fractions with numerator 1 and denominators 1 or $2\sqrt{5}$, and the square root expression $\sqrt{5}$. 3. **Simplifying the fractions:** - $\frac{1}{1} = 1$ - $\frac{1}{2\sqrt{5}}$ can be rationalized: $$\frac{1}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{2 \times 5} = \frac{\sqrt{5}}{10}$$ 4. **Interpreting the layout:** The sequence seems to be a pattern of numbers decreasing diagonally, possibly representing terms in a series or matrix elements. 5. **Summary:** The key values are 2, 1, and $\frac{\sqrt{5}}{10}$, arranged in a stepped diagonal pattern. Final simplified key values: $$2, \quad 1, \quad \frac{\sqrt{5}}{10}$$