Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem28

Video Transcription

Question number 28. The square root here of 0.4444, where the 4 repeats 2010 times, was written as an infinite decimal in base 10.  What is the value of the digit with subscript equal to 2010? Here we can use the methods of the calculus and write a linearization of our number.  So we recall from the calculus that 1 plus here quantity to the power n can be approximated by a line like so for values of x near 0.  So that's a pretty good approximation. So we can use that. So here n is equal to 1 half and x would be equal to negative 0.5555 dot dot dot and the last digit here is a 6.  So that's one way to do this but this could be sort of a tedious calculation. So to avoid being tedious what we can do is reverse engineer the answer and we also we can check.  So for example looking at 0.666 squared may be a little easier 0.66 squared may be a little easier than that 0.6 squared if we were to just build the answer and square it to see how close do we end up near 0.444  we would have that the approximation here in the middle gives us 0.4354 obviously here we get 0.36 and here we would have 0.443556  and so we notice that the more 6s repeat the more 4s here appear in the front so with 2010 repeating 4s the best answer here would be if we had to guess that the digit with index 2010 is equal to 6.  And so here this question is a little too tedious to compute directly so we just make our best guess here and choose 6.

Video Summary

Summary Not Available