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The Olympiad task For the 10th grade if you do not know how to solve it immediately sayCorrect answer 2

The Olympiad task For the 10th grade if you do not know how to solve it immediately-example-1
User Krishna Kant Sharma
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1 Answer

17 votes
17 votes

Given:-


f((1)/(x+1))=(1)/(2)f(x),f(1-x)=1-f(x)

To find:-


s_n=f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))

Now we simplify above equation to solve it,


\begin{gathered} f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))=f(1)+f((2-1)/(2))+f((3-2)/(3))+\cdots+f((n+1-n)/(n)) \\ f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))=f(1)+f(1-(1)/(2))+f(1-(2)/(3))+\ldots+f(1-((n-1))/(n)) \\ f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))=f(1)+1-f((1)/(2))+1-f((2)/(3))+\cdots+1-f((n-1)/(n)) \\ f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))=f(1)-f((1)/(2))-f((2)/(3))-\cdots-f((n-1)/(n))+1(n-1) \\ f(1)+f((1)/(2))+f((1)/(3))+\cdots+f((1)/(n))=f(1)-f((1)/(1+1))-f((2)/(2+1))-\cdots-f((n-1)/((n-1)+1))+1(n-1) \end{gathered}

By furthur simplification we get,


\begin{gathered} f(1)-f((1)/(1+1))-f((2)/(2+1))-\cdots-f((n-1)/((n-1)+1))+1(n-1)=f(1)-(1)/(2)f(1)-(1)/(2)f(2)-\cdots-(1)/(2)f(n-1)+1(n-1)_{} \\ \text{ =}-(1)/(2)f(1)-(1)/(2)f(2)-\cdots-(1)/(2)f(n-1)+1(n-1) \\ \text{ =-}(1)/(2)\lbrack f(1)+f(2)+...+f(n-1)\rbrack+1(n-1) \end{gathered}

So the requiired answer is,


\text{-}(1)/(2)\lbrack f(1)+f(2)+...+f(n-1)\rbrack+1(n-1)

User Leonela
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