Let $n$ be a positive integer, such that the coefficients of the fifth, sixth and seventh terms, with respect to $x$, of the development of $\bigg (\frac {log_n (\sqrt {2 ^ n})} {log_e (n) .log_n (\sqrt {2 ^ e)}} + x \bigg) ^ n$ according to the decreasing powers of $x$, they are in arithmetic progression. Determine $n$.
We have that $n\geq6$
Using the infomation given, I wrote $2\binom{n}{5}\cdot\bigg(\frac{n\log_ne}{e}\bigg)^5=\binom{n}{4}\cdot\bigg(\frac{n\log_ne}{e}\bigg)^4+\binom{n}{6}\cdot\bigg(\frac{n\log_ne}{e}\bigg)^6$
Simplifying this, I got $n^2(n-4)(n-5)\log_ne-6n(n-4)e\log_ne+3e=0$
I don't know what I have to do now :/
Can someone help me?
Thanks for attention.