The value of x that satisfies the relation x | vedclass

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(C) The given expression is an infinite geometric series: $x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dots \infty$.This is a geometric series with the first

Vedclass · Accepted Answer

$2 \sin 18^\circ$

Vedclass · Answer

(C) The given expression is an infinite geometric series: $x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dots \infty$.This is a geometric series with the first term $a = 1$ and common ratio $r = -x$.The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$,provided $|r| Substituting the values,we get $x = \frac{1}{1 - (-x)} = \frac{1}{1 + x}$.This simplifies to $x(1 + x) = 1$,which is $x^2 + x - 1 = 0$.Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,we get $x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2}$.Since the series must converge,we require $|-x| The value $\frac{-1 - \sqrt{5}}{2} \approx -1.618$ is rejected as it is less than $-1$.Thus,$x = \frac{\sqrt{5} - 1}{2}$.We know that $\sin 18^\circ = \frac{\sqrt{5} - 1}{4}$.Therefore,$x = 2 \sin 18^\circ$.

The value of $x$ that satisfies the relation $x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dots \infty$.

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