nC2=1/4[n+2C2]
If by "nC2", you mean " Combinations of n objects 2 at a time", then there is no " natural number n" solution! It looks like something you made up on the fly!!.
+9665 \(nC2=\frac{n+2C2}{4}\)
\(4(\frac{n!}{(2)[(n-2)!]})=n+1\)
\(2n(n-1)=n+1\)
\(2n^2-2n=n+1\)
\(2n^2-3n-1=0\)
Using quad. formula,
\(n = {-(-3) \pm \sqrt{(-3)^2-4(2)(-1)} \over 2(2)}\)
\(n=\frac{3+\sqrt{17}}{4}\) or \(n=\frac{3-\sqrt{17}}{4}\)
There are no natural number solutions but some real number solutions.
+9665 Or you mean \(nC2=\frac{1}{4[n+2C2]}\)?
I will solve this one for you
\(nC2=\frac{1}{4(n+2C2)}\)
\(4(n+1)=\frac{((2!)(n-2)!)}{n!}\)
\(4(n+1)=\frac{2}{(n)(n-1)}\)
\(2(n+1)(n-1)(n)=1\)
\(2n^3-2n-1=0\)
We will use the cubic equation.
First \(\Delta_0=0^2-3(2)(-2)=0+12=12\)
Then\(\Delta_1=2(0)^3-9(2)(0)(-2)+27(2)^2(-1)=0+0-108=-108\)
\(C=\sqrt[3]{\frac{\sqrt{(\Delta_1)^2-4(\Delta_0)^3+\Delta_1}}{2}}=\sqrt[3]{\frac{\sqrt{(-108)^2-4(12)^3-108}}{2}}=\frac{\sqrt[6]{4644}}{\sqrt[3]{2}}\)
\(u=\frac{-1+\sqrt3i}{2}\)
\(u^1C=\frac{(\sqrt{3}i-1)(\sqrt[6]{4644})}{\sqrt[3]{16}}\)
\(u^2C=(\frac{\sqrt3i-1}{2})^2(\frac{\sqrt[6]{4644}}{\sqrt[3]{2}})=(\frac{-1-\sqrt3i}{2})(\frac{\sqrt[6]{4644}}{\sqrt[3]{2}})=-\frac{(1+\sqrt3i)(\sqrt[6]{4644})}{\sqrt[3]{16}}\)
\(u^3C=((\frac{-1+\sqrt3i}{2})^3)(\frac{\sqrt[6]{4644}}{\sqrt[3]{2}})=\frac{\sqrt[6]{4644}}{\sqrt[3]{2}}\)
\(n_1=\frac{0+\frac{(\sqrt3i-1)(\sqrt[6]{4644})}{\sqrt[3]{16}}+\frac{12}{\frac{(\sqrt3i-1)(\sqrt[6]{4644})}{\sqrt[3]{16}}}}{6}\)
This is definitely not a natural number XD
\(n_2=\frac{0-\frac{(1+\sqrt3i)(\sqrt[6]{4644})}{\sqrt[3]{16}}+\frac{12}{\frac{(1+\sqrt3i)(\sqrt[6]{4644})}{\sqrt[3]{16}}}}{6}\)
This is still not a natural number XD
\(n_3=\frac{\frac{\sqrt[6]{4644}}{\sqrt[3]{2}}+\frac{12}{\frac{\sqrt[6]{4644}}{\sqrt[3]{2}}}}{6}\)
This is not a natural number :(
LOL This no natural solution.
Sorry i have gone a bit crazy solving that cubic eq.
