Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 218399 by Nicholas666 last updated on 09/Apr/25

Answered by MrGaster last updated on 10/Apr/25

determinant (((f(n)=(n^2 /2))))when n is even f(n)=(n^2 /2) f(n)=((n^2 −τ(n))/2) f(n)+f(f(n))=((n^2 −τ(n))/2)+(((((n^2 −τ(n))/2) )^2 −τ(((n^2 −τ(n))/2)))/2) =((n^2 −τ(n))/2)+((n^4 −2n^2 τ(n)+τ(n)^2 −4τ(((n^2 −τ(n))/2)))/8) does not hold f(n)+f(f(n))=τ(n)n+τ(τ(n)n)∙τ(n)n=n^2 ⇒τ(n)(n+τ(τ(n)n)∙τ(n)n)=n^2 no solution determinant (((f(n)=n^2 −τ(n)^2 ))) f(n)+f(f(n))=n^2 −τ(n)^2 +(n^2 −τ(n)^2 )^2 −τ(n^2 −τ(n)^2 )^2 =n^2 does not hold determinant (((f(n)=(n^2 /(1+τ(n)))))) ⇒(n^2 /(1+τ(n)))+((((n^2 /(1+τ(n))))^2 )/(1+τ((n^2 /(1+τ(n))))))=n^2 does not hold determinant (((f(n)=(n^2 /2) when τ(n)is even,f(n)=((n^2 −1)/2)when τ(n)is odd))) verify n=2(τ=2even):f(2)=2,f(2)+f(2)=4=2^2 n=4(τ=3odd):f(4)=((16−1)/2)=7.5 not an integer determinant (((f(n)= { (((n^2 /2) n evem)),((((n^2 −1)/2) n odd)) :}))) n=3:f(3)=4,f(4)=8⇒4+8=12≠9 error determinant (((f(n)=(n^2 /2) ∀n∈Z^+ )))but only when n=2 holds final solution must satisfy f(n)+f(f(n))=n^2 and f(n)is related to τ(n) determinant (((f(n)=n^2 −τ(n))))partially holds e.g.,n=2,f(2)=2,f(f(2))=2 ⇒2+2=4=2^2

$$\begin{array}{|c|}{{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} }{\mathrm{2}}}\\\hline\end{array}\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{even} \\ $$$$\cancel{{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} }{\mathrm{2}}} \\ $$$${f}\left({n}\right)=\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}} \\ $$$${f}\left({n}\right)+{f}\left({f}\left({n}\right)\right)=\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}}+\frac{\left(\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}}\:\right)^{\mathrm{2}} −\tau\left(\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}}\right)}{\mathrm{2}} \\ $$$$=\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}}+\frac{{n}^{\mathrm{4}} −\mathrm{2}{n}^{\mathrm{2}} \tau\left({n}\right)+\tau\left({n}\right)^{\mathrm{2}} −\mathrm{4}\tau\left(\frac{{n}^{\mathrm{2}} −\tau\left({n}\right)}{\mathrm{2}}\right)}{\mathrm{8}} \\ $$$$\cancel{\mathrm{does}\:\mathrm{not}\:\mathrm{hold}} \\ $$$${f}\left({n}\right)+{f}\left({f}\left({n}\right)\right)=\tau\left({n}\right){n}+\tau\left(\tau\left({n}\right){n}\right)\centerdot\tau\left({n}\right){n}={n}^{\mathrm{2}} \\ $$$$\Rightarrow\tau\left({n}\right)\left({n}+\tau\left(\tau\left({n}\right){n}\right)\centerdot\tau\left({n}\right){n}\right)={n}^{\mathrm{2}} \\ $$$$\cancel{\mathrm{no}\:\mathrm{solution}} \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)={n}^{\mathrm{2}} −\tau\left({n}\right)^{\mathrm{2}} }\\\hline\end{array} \\ $$$${f}\left({n}\right)+{f}\left({f}\left({n}\right)\right)={n}^{\mathrm{2}} −\tau\left({n}\right)^{\mathrm{2}} +\left({n}^{\mathrm{2}} −\tau\left({n}\right)^{\mathrm{2}} \right)^{\mathrm{2}} −\tau\left({n}^{\mathrm{2}} −\tau\left({n}\right)^{\mathrm{2}} \right)^{\mathrm{2}} ={n}^{\mathrm{2}} \\ $$$$\cancel{\mathrm{does}\:\mathrm{not}\:\mathrm{hold}} \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} }{\mathrm{1}+\tau\left({n}\right)}}\\\hline\end{array} \\ $$$$\Rightarrow\frac{{n}^{\mathrm{2}} }{\mathrm{1}+\tau\left({n}\right)}+\frac{\left(\frac{{n}^{\mathrm{2}} }{\mathrm{1}+\tau\left({n}\right)}\right)^{\mathrm{2}} }{\mathrm{1}+\tau\left(\frac{{n}^{\mathrm{2}} }{\mathrm{1}+\tau\left({n}\right)}\right)}={n}^{\mathrm{2}} \\ $$$$\cancel{\mathrm{does}\:\mathrm{not}\:\mathrm{hold}} \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\mathrm{when}\:\tau\left({n}\right)\mathrm{is}\:\mathrm{even},{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}}\mathrm{when}\:\tau\left({n}\right)\mathrm{is}\:\mathrm{odd}}\\\hline\end{array} \\ $$$$\mathrm{verify}\:{n}=\mathrm{2}\left(\tau=\mathrm{2even}\right):{f}\left(\mathrm{2}\right)=\mathrm{2},{f}\left(\mathrm{2}\right)+{f}\left(\mathrm{2}\right)=\mathrm{4}=\mathrm{2}^{\mathrm{2}} \\ $$$${n}=\mathrm{4}\left(\tau=\mathrm{3odd}\right):{f}\left(\mathrm{4}\right)=\frac{\mathrm{16}−\mathrm{1}}{\mathrm{2}}=\mathrm{7}.\mathrm{5}\:\cancel{\mathrm{not}\:\mathrm{an}\:\mathrm{integer}} \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)=\begin{cases}{\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:{n}\:\mathrm{evem}}\\{\frac{{n}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}}\:{n}\:{odd}}\end{cases}}\\\hline\end{array} \\ $$$${n}=\mathrm{3}:{f}\left(\mathrm{3}\right)=\mathrm{4},{f}\left(\mathrm{4}\right)=\mathrm{8}\Rightarrow\mathrm{4}+\mathrm{8}=\mathrm{12}\neq\mathrm{9}\:\cancel{\mathrm{error}} \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)=\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\forall{n}\in\mathbb{Z}^{+} }\\\hline\end{array}\mathrm{but}\:\mathrm{only}\:\mathrm{when}\:{n}=\mathrm{2}\:\mathrm{holds} \\ $$$$\mathrm{final}\:\mathrm{solution}\:\mathrm{must}\:\mathrm{satisfy}\:{f}\left({n}\right)+{f}\left({f}\left({n}\right)\right)={n}^{\mathrm{2}} \:\mathrm{and}\:{f}\left({n}\right)\mathrm{is}\:\mathrm{related}\:\mathrm{to}\:\tau\left({n}\right) \\ $$$$\begin{array}{|c|}{{f}\left({n}\right)={n}^{\mathrm{2}} −\tau\left({n}\right)}\\\hline\end{array}\mathrm{partially}\:\mathrm{holds}\:\mathrm{e}.\mathrm{g}.,{n}=\mathrm{2},{f}\left(\mathrm{2}\right)=\mathrm{2},{f}\left({f}\left(\mathrm{2}\right)\right)=\mathrm{2}\:\Rightarrow\mathrm{2}+\mathrm{2}=\mathrm{4}=\mathrm{2}^{\mathrm{2}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com