$\"\"$ \ \

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The value of $\"\"$ .

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(a)

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$\"\"$

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Multiply each side by $\"\"$ .

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$\"\"$

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$\"\"$ \ \

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Threfore, the value is $\"\"$ .

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$\"\"$

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(b) Find the sum of the geometric series for the value of the $\"\"$ .

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Expand the value $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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The general form of geometric series is $\"\"$ .

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Comapre the expression with general form.

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The initial term is $\"\"$ and the common ratio is $\"\"$ .

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The sum of the $\"\"$ terms is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$ .

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Therefore, the sum of the series is $\"\"$ .

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$\"\"$

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(c) Find the number of decimal representations of the integer number $\"\"$ .

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The value is $\"\"$ .

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Rewrite the value .

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$\"\"$

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The value of $\"\"$ .

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The valus of $\"\"$ and $\"\"$ are the same.

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Therefore, the number one has $\"\"$ number of decimal representations.

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$\"\"$

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(d) Find the numbers have more than one decimal representations.

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Except the number $\"\"$ , all the rational numbers have more than one decimal representation.

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Hence all the rational numbers with a terminating decimal representation, except $\"\"$ .

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$\"\"$

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(a) The value is $\"\"$ .

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(b) The sum of the series is $\"\"$ .

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(c) The number one has $\"\"$ number of decimal representations.

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(d) Except the number $\"\"$ , all the rational numbers have more than one decimal representation.