Write the repeating decimal as a geometric series.

Question

Write the repeating decimal  as a geometric series. ​

Quizplus · Accepted Answer

To convert the repeating decimal to a geometric series, we need to first write it in the form:
a + ar + ar^2 + ar^3 + ...

where a is the first term and r is the common ratio.

Starting from the end of the repeating block and working backwards, we can see that the last group of digits before the "TB7497_11" is "1af7". This can be written as:
1af7 / 10^4

Moving to the left, the next group of digits is "83f9b729". We can add this to the previous fraction by multiplying it by the common ratio, which is 1/10^8 since there are 8 digits between the two groups:
(1af7 / 10^4) + (83f9b729 / 10^12)

Continuing this process, we can add the following groups of digits to the previous sum by multiplying by the common ratio:
_e15b_bedf (common ratio = 1/10^4)
_ec46 (common ratio = 1/10^4)
_11 (common ratio = 1/10^2 since there are 2 digits between this group and the previous)

Putting this all together, we get:
(1af7 / 10^4) + (83f9b729 / 10^12) + (_e15b_bedf / 10^16) + (_ec46 / 10^20) + (_11 / 10^22) + ...

This can be simplified as:
(1af7 / 10^4) / (1 - 1/10^4) = 1af7 / 9999 * 10^4
+ (83f9b729 / 10^12) / (1 - 1/10^4) = 83f9b729 / 9999 * 10^8
+ (_e15b_bedf / 10^16) / (1 - 1/10^4) = e15b_bedf / 9999 * 10^12
+ (_ec46 / 10^20) / (1 - 1/10^4) = ec46 / 9999 * 10^16
+ (_11 / 10^22) / (1 - 1/10^2) = 11 / 99 * 10^22
+ ...

Therefore, the repeating decimal 11eac44f_ec46_e15b_bedf_83f9b7291af7_TB7497_11 can be written as the geometric series:
(1af7 / 9999 * 10^4) + (83f9b729 / 9999 * 10^8) + (e15b_bedf / 9999 * 10^12) + (ec46 / 9999 * 10^16) + (11 / 99 * 10^22) + ...

Write the Repeating Decimal as a Geometric Series