Question 1122513
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(a) x = 0.*7* = 7/10 + 7/100 + 7/1000 + ...<br>
This is a geometric progression with first term 7/10 and common ratio 1/10; the infinite sum is<br>
{{{(7/10)/(1-(1/10)) = (7/10)/(9/10) = 7/9}}}<br>
Answer: 0.*7* = 7/9<br>
(b) x = 0.3*15* = 3/10 + 15/1000 + 15/100000 + 15/10000000 + ...<br>
After the first term, the rest is a geometric progression with first term 15/1000 and common ratio 1/100; its infinite sum is<br>
{{{(15/1000)/(1-(1/100)) =  (15/1000)/(99/100) = (15/1000)/(990/1000) = 15/990}}}<br>
Then the entire decimal number is<br>
{{{3/10 + 15/990 = 297/990 + 15/990 = 312/990}}}<br>
Answer: 0.3*15* = 312/990; simplify if required.