ADVANCED EXERCISES - Decimal Expansions

(1)
Show that a + ar + ar2 + ar3 + ... + arn 
          1 - rn + 1
     = a ------- .
          1 - r
( Click On HINT To Get Some Help , Click On SOLUTION To See The Answer )


(2)
If -1 < r < 1, then the infinite sum a + ar + ar2 + ar3 + ... has meaning as the number that the finite sum a + ar + ar2 + ar3 + ... + arn approaches as n gets arbitrarily large. What is this number?
( HINT , SOLUTION )


(3)
Express the following repeating decimals in the form a + ar + ar2 + ar3 + ... .
               _
        (i)  0.7
               __
       (ii)  0.32
Use Exercise (2) to convert each to rational form.
( HINT , SOLUTION )


(4)
Show that SQRT(2) CANNOT be written in a rational form, or equivalently, that SQRT(2) cannot be written in the form p/q, where p and q are integers.
( HINT , SOLUTION )


(5)
Can the number 0.110100100010000100000... be written in a rational form?
( HINT , SOLUTION )


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