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What is the common ratio of the G.P. \((\sqrt{10} + \sqrt{5}) + (\sqrt{10} + 2\sqrt{5}) + ... \)?...


Question

What is the common ratio of the G.P. \((\sqrt{10} + \sqrt{5}) + (\sqrt{10} + 2\sqrt{5}) + ... \)?

Options

A) \(\sqrt{2}\)

B) \(\sqrt{5}\)

C) 3

D) 5

The correct answer is A.

Explanation:

Common ratio r of the G.P is
\(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)
\( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)
\(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)
\(\frac{\sqrt{50}}{5}\)
\(\frac{\sqrt{25 \times 2}}{5}\)
\(\frac{5\sqrt{2}}{5}\)
\(\sqrt{2}\)

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Dicussion (1)

  • Common ratio r of the G.P is
    \(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)
    \(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)
    \(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)
    \( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)
    \(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)
    \(\frac{\sqrt{50}}{5}\)
    \(\frac{\sqrt{25 \times 2}}{5}\)
    \(\frac{5\sqrt{2}}{5}\)
    \(\sqrt{2}\)

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