Answer:
the final answer is 50/11
Step-by-step explanation:
Here let's regard the first number of this geometric series as 0.54, holding the 4 to include later. The next is 0.0054, the next 0.000054, and so on.
Thus, the common ratio is 1/100. Then the sum of the infinite series, not including that 4, is
a 0.54 0.54
-------- = ------------ = ------------
1 - r 1 - 1/100 99/100
54
Multiplying both 0.54 and 99/100 by 100 results in ------- and this
99 0.545454....
Now add the 4 back in, obtaining 4 54/99, or (396 + 54) / 99.
This is the same as 450 / 99. You can readily check with a calculator to see whether this is equivalent to the given 4.54545454545...
Note that 450/99 is in the form p/q (not pq), where p and q are positive integers. But also note that 450/99 can be reduced to 150/33, or
50/11. A calculator will show you that 50/11 is equivalent to the given 4.54545454545...
Hence, the final answer is 50/11 (in the form p/q, NOT pq).
