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Maths-

General

Easy

Question

The rational number which equals the number $Error converting from MathML to accessible text.$ with recurring decimal is

2001 (
$\frac{2379}{997}$
0
None of these

The correct answer is: $fraction numerator 2379 over denominator 997 end fraction$

$negative 2001$
$b subscript 3 end subscript equals fraction numerator 1 over denominator 1 minus b subscript 2 end subscript end fraction equals fraction numerator 1 over denominator 1 minus fraction numerator 1 over denominator 1 minus b subscript 1 end subscript end fraction end fraction equals fraction numerator 1 minus b subscript 1 end subscript over denominator negative b subscript 1 end subscript end fraction equals fraction numerator b subscript 1 end subscript minus 1 over denominator b subscript 1 end subscript end fraction$
$b subscript 1 end subscript equals b subscript 3 end subscript blank rightwards double arrow blank b subscript 1 end subscript superscript 2 end superscript minus b subscript 1 end subscript plus 1 equals 0$
$rightwards double arrow b subscript 1 end subscript equals negative omega blank o r blank omega to the power of 2 end exponent rightwards double arrow b subscript 2 end subscript equals fraction numerator 1 over denominator 1 plus omega end fraction equals negative omega blank o r blank omega to the power of 2 end exponent$
$not stretchy sum from r equals 1 to 2001 of b subscript r end subscript superscript 2001 end superscript equals not stretchy sum from r equals 1 to 2001 of open parentheses negative omega close parentheses to the power of 2001 end exponent$
$equals negative not stretchy sum from r equals 1 to 2001 of 1$
$equals negative 2001$
$equals 2 plus fraction numerator 357 over denominator 10 to the power of 3 end exponent end fraction plus fraction numerator 357 over denominator 10 to the power of 6 end exponent end fraction plus... infinity$
$equals 2 plus fraction numerator fraction numerator 357 over denominator 10 to the power of 3 end exponent end fraction over denominator 1 minus fraction numerator 1 over denominator 10 to the power of 3 end exponent end fraction end fraction$
$equals 2 plus fraction numerator 357 over denominator 999 end fraction equals fraction numerator 2355 over denominator 999 end fraction$
Alternative solution:
Let,
$Error converting from MathML to accessible text.$
$Error converting from MathML to accessible text.$
On subtracting, we get
$Error converting from MathML to accessible text.$

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