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

MathematicsIf x=7+1+7+1+7⋯, then find the value of (x4−14×2−x)

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

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

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

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

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

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

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

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

$∴ x^{4} - 14 x^{2} - x = - 48$