Science:Math Exam Resources/Courses/MATH101/April 2012/Question 01 (h)

MATH101 April 2012

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Question 01 (h)
Short-Answer Question. Show all your work, simplify your answer as much as possible. Express $2.65656565....$ as a rational number i.e. in the form $p/q$ where p and q are integers.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try to write this as a sum of fractions. Then use facts about geometric series to help.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Notice that we have $2.656565....=2+{\frac {65}{100}}+{\frac {65}{10000}}+{\frac {65}{1000000}}=2+\sum _{n=1}^{\infty }{\frac {65}{100^{n}}}$ Now, recall that the sum of a geometric series is $\sum _{n=1}^{\infty }ar^{n-1}={\frac {a}{1-r}}.$ In our case, we have that $a=65/100$ (the term when we plug in $n=1$ into the sum) and we have that $r=1/100$ and hence $2+\sum _{n=1}^{\infty }{\frac {65}{100^{n}}}=2+{\frac {\tfrac {65}{100}}{1-{\tfrac {1}{100}}}}=2+{\frac {65}{99}}={\frac {263}{99}}$ completing the question.

Solution 2
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $x=2.65656565656565....$ Notice that $100x=265.656565656565...$ Subtracting the two equations gives $99x=263$ and so $x={\frac {263}{99}}$

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