Marinada '24

[lang=hr] Suma $$\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\ldots +\sqrt{1+\dfrac{1}{20223^2}+\dfrac{1}{20224^2}}$$ je broj oblika $p/q$ za $p,q$ prirodne i relativno proste. Odredite $p+q$. [/lang] [lang=en] The sum $$\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\ldots +\sqrt{1+\dfrac{1}{20223^2}+\dfrac{1}{20224^2}}$$ is a number that can be expressed in the form $p/q$, where $p$ and $q$ are natural numbers that are relatively prime. Find $p+q$. [/lang]