Fractions in a row: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11. Prove that the signs + and - cannot be inserted between so the result is 0
The following fractions are written in a row:
1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11.
Prove that one cannot insert the signs + and - between these numbers so that the result would be 0.
If any information/working/answers given could please be explained in simple terms. Thank you very much! (:
2 Answers | Add Yours
without any sign thats not possible
I think this should be able to answer your question. The first thing we must do is convert all of these into fractions (meaning convert 1 into its fraction) as follows:
1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11
Now, we must apply the rule of addition/subtration of fractions, which means we must find the lowest common denominator for all of these fractions. After we find all the factors and do the multiplication, we should find the answer. For (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) the least common multiple (LCM) is 27720. Therefore, the lowest common denominator (LCD) is 27720. Now we multiply each of the numerators by what it took to reach our denominator. We then rewrite the equivilant fractions with the new LCD as follows:
27720/27720, 13860/27720, 9240/27720, 6930/27720, 5544/27720, 4620/27720, 3960/27720, 3465/27720, 3080/27720, 2772/27720, 2520/27720
Now, no matter how hard I try mixing or match with adding here and subtracting there, I will never get the sums/differences of these fractions to equal zero. Why? Because two of them just foul up the whole program. 5544/27720 and 2772/27720. Because these do not end with a 5 or a 0, they will throw off any results regardless of how we try to order them.
Hope this helps!
Join to answer this question
Join a community of thousands of dedicated teachers and students.Join eNotes