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If the sum of first 10 terms of $\log_{5} x + \log_{5^{1 / 2}} x + \log_{5^{1 / 3}} x + - - - - -$ is 550 and $S_{n}$ is sum of ‘n’ terms then the quadratic equation whose roots are $\log_{10} S_{1}$ and $\log_{10} S_{4}$ is

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$x^{2} + 3 x + 2 = 0$

$x^{2} + 2 x + 3 = 0$

$x^{2} - 3 x + 2 = 0$

$x^{2} - 2 x + 3 = 0$

answer is C.

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Detailed Solution

$\log_{5}^{x} + \log_{5}^{x^{2}} + \log_{5}^{x^{3}} + - - - = 550$

$\log_{5} [x . x^{2} . x^{3} - - - - - t e r m s] = 550$

$\log_{5} [x^{55}] = 550 \Rightarrow (55) \log_{5}^{x} = 550$

$x = 5^{10}$

$S_{n} = \log_{5}^{x} [1 + 2 + ..... + n] = \frac{n (n + 1)}{2} \log_{5}^{x} = 5 n (n + 1)$

$∴ s_{1} = 10, s_{4} = 100$

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