Walking on logs

Calculus Level 5

A man begins his journey at the origin \( (0,0)\) of the Cartesian plane. He then walks \(\ln\left(\frac{8}{4}\right)\) units right (positive \(x\) direction), \(\ln\left(\frac{9}{5}\right)\) units up (positive y direction), \(\ln\left(\frac{10}{6}\right)\) units left (negative \(x\) direction), \(\ln\left(\frac{11}{7}\right)\) units down (negative \(y\) direction).

He continues this pattern indefinitely with the \(n\)-th side of this spiral being of length \(\ln\left(\frac{n+7}{n+3}\right)\) starting with \(n=1\).

Find the positive distance (magnitude) that the man will travel from the origin. Give your answer to 2 decimal places.

Inspired by my classmate Kaishu Mason


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