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Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
(a1/2+2a+2a1/2+1a1/22a1)a1/2+1a1/2

Solution
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(a12+2a+2a12+1a122a1)a12+1a12
=(a+2(a)2+2a+1)a2(a)21))a+1a
=[a+2(a+1)2a2(a+1)(a1)]a+1a
=[a+2a+1a2a1]a+1a(a+1)
=((a+2)(a1)(a2)(a+1)a1)1a
=[aa+2a2(a+a2a2)a1]1a
=[a+a2a+a+2a1]1a
=2a(a1)a=2a1

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