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Exploring the Common Ratio in Geometric Sequences with Unique Sum Conditions

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When we dive into the world of geometric sequences, one of the most intriguing questions revolves around the value of the common ratio. This article aims to explore the scenario where the sum to infinity of a geometric sequence is twice the sum of the first two terms. We will utilize mathematical formulas and properties to delve into this problem.

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Mathematical Foundations and Sum to Infinity Formula

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Let's begin by revisiting the sum to infinity formula for a geometric sequence. Given a geometric sequence with the first term (a) and common ratio (r), the sum to infinity (S_{infty}) is defined as:

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$S_{infty} frac{a}{1 - r}$

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Under the condition that (|r| " "

Deriving the Common Ratio

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Let's denote the first term by (a) and the common ratio by (r). The sum of the first two terms of the geometric sequence is:

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(S_{2} a ar a(1 r))

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Given that the sum to infinity is twice this value, we can write:

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$frac{a}{1 - r} 2a(1 r)$

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We can simplify this equation by dividing both sides by (a), assuming (a eq 0):

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$frac{1}{1 - r} 2(1 r)$

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Expanding and rearranging gives us:

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$frac{1}{1 - r} 2 2r$

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To clear the fraction, we multiply both sides by (1 - r):

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$1 (2 2r)(1 - r)$

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Simplifying further:

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$1 2 - 2r 2r - 2r^2$

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$1 2 - 2r^2$

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$2r^2 1$

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$r^2 frac{1}{2}$

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$r pm sqrt{frac{1}{2}}$

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This gives us two possible values for (r):

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$r_1 sqrt{frac{1}{2}}$

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$r_2 -sqrt{frac{1}{2}}$

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Conclusion and Verification

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In conclusion, the possible values of the common ratio (r) that satisfy the condition that the sum to infinity of a geometric sequence is twice the sum of the first two terms are:

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$r frac{sqrt{2}}{2}$ or $r -frac{sqrt{2}}{2}$

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It is worthwhile to verify that these values meet the initial condition:

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$S_{infty} frac{a}{1 - frac{sqrt{2}}{2}} 2a(1 frac{sqrt{2}}{2}) 2a sqrt{2}$

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$S_{infty} frac{a}{1 frac{sqrt{2}}{2}} 2a(1 - frac{sqrt{2}}{2}) 2a - sqrt{2}$

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Both conditions hold true, confirming our solution. Understanding these concepts is fundamental to working with geometric sequences and their properties.