What is 5280th digit in Decimal Expansion of 9/17?

Solution:

From the definition of a rational number, 9/17 is a rational number.

So, decimal value of 9/17 must be either a terminating decimal or a recurring decimal.

Value of 9/17 is approximately equal to 0.52941176470588235294117647058824……..

We can observe that the decimal is non-terminating. So, it must be a recurring decimal. Now, we have to identify, after how many places the pattern is repeated. We can notice that the repeated pattern is very long.

So, let us multiply the given number by a large power of ten so that the place of the decimal point shifts to the right. Here, we multiply the number by 10¹⁰.

0.52941176470588235294117647058824… × 10000000000 = 5294117647.0588235294117647058824…

Now, let us subtract the whole number part from the obtained number.

5294117647.05882352941176470588235……. − 5294117647 = 0.05882352941176470588235294..……

We can notice a repeating pattern a bit farther in the obtained decimal expansion. Now, repeat the same technique that we applied above, i.e., multiplying the given number by a large power of ten and then subtracting. 

0.0582352941176470588235294…….. × 1000000 = 0.52941176470588235294117647058824……..

Now, we can observe the repeat pattern in full. By counting carefully, we can observe that the pattern is repeating after 16 digits (5294117647058823). 

Now, we need to find the 5280th digit in the decimal expansion of 9/17. Divide 5280 by 16.

5280 ÷ 16 = 330

So, 5280 is a multiple of 16, that means the pattern will be completed for the 330th time on the 5280th digit. As we know, the end digit of the pattern is 3, so the 5280th digit will be 3.

Hence, 5280th digit in the decimal expansion of 9/17 is 3.

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5280th digit in Decimal Expansion of 9/17 is 3 and the detailed solution for the same is added below.