If x=7+1+7+1+7⋯, then find the value of (x4−14x2−x)

x=7+1+7+1+7x=7+1+xx2=7+1+xx2−7=1+x(x2−7)2=1+xx4−14x2+49=1+xx4−14x2−x=1−49=−48∴x4−14x2−x=−48

If x=7+1+7+1+7⋯, then find the value of (x4−14x2−x)

x=7+1+7+1+7x=7+1+xx2=7+1+xx2−7=1+x(x2−7)2=1+xx4−14x2+49=1+xx4−14x2−x=1−49=−48∴x4−14x2−x=−48

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If $x = \sqrt{7 + \sqrt{1 + \sqrt{7 + \sqrt{1 + \sqrt{7}}}}} \dots$ , then find the value of $(x^{4} - 14 x^{2} - x)$

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$- 48$

$49$

$- 49$

answer is B.

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Detailed Solution

$x = \sqrt{7 + \sqrt{1 + \sqrt{7 + \sqrt{1 + \sqrt{7}}}}}$

$x = \sqrt{7 + \sqrt{1 + x}}$

$x^{2} = 7 + \sqrt{1 + x}$

$x^{2} - 7 = \sqrt{1 + x}$

${(x^{2} - 7)}^{2} = 1 + x$

$x^{4} - 14 x^{2} + 49 = 1 + x$

$x^{4} - 14 x^{2} - x = 1 - 49 = - 48$

$∴ x^{4} - 14 x^{2} - x = - 48$

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