Let $f$ be a polynomial function such that $\log_2(f(x)) = \left(\log_2\left(2+\dfrac{2}{3}+\dfrac{2}{9}+\ldots\infty\right)\right)\cdot\log_3\left(1+\dfrac{f(x)}{f(1/x)}\right)$, $x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to ________.

The sum of the infinite geometric progression is given by:

$S = 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ldots \infty = \dfrac{2}{1 - \dfrac{1}{3}} = 3$

Substituting this into the given equation:

$\log_2(f(x)) = \log_2(3) \cdot \log_3\left(1 + \dfrac{f(x)}{f(1/x)}\right)$

Using the base change property $\log_a(b) \cdot \log_b(c) = \log_a(c)$:

$\log_2(f(x)) = \log_2\left(1 + \dfrac{f(x)}{f(1/x)}\right)$

Equating the arguments:

$f(x) = 1 + \dfrac{f(x)}{f(1/x)}$

$f(x)f(1/x) - f(1/x) = f(x)$

$f(x)f(1/x) = f(x) + f(1/x)$

The only polynomial functions satisfying this relation are of the form $f(x) = 1 \pm x^n$.

Given $f(6) = 37$:

$1 \pm 6^n = 37$

Taking the positive sign, $6^n = 36 \Rightarrow n = 2$.

Thus, the polynomial is $f(x) = x^2 + 1$.

The required sum is:

$\sum_{n=1}^{10} f(n) = \sum_{n=1}^{10} (n^2 + 1)$

$\sum_{n=1}^{10} f(n) = \sum_{n=1}^{10} n^2 + \sum_{n=1}^{10} 1$

$\sum_{n=1}^{10} f(n) = \dfrac{10 \times 11 \times 21}{6} + 10$

$\sum_{n=1}^{10} f(n) = 385 + 10 = 395$

Answer: $395$