FMM18

Friday Morning Math Problem

Radical Sums

Evaluate the sum:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dfrac{1}{1 + \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dfrac{1}{\sqrt{3} + \sqrt{4}} + \dots + \dfrac{1}{\sqrt{1977} + \sqrt{1978}} }

Solution

There are 1977 terms to add, and surely it would not be a Friday Morning Math Problem if it were so boring as to tediously add so many terms. You'll also notice the radicals are improper. If we try getting rid of the radicals on the bottom of each term, we get something interesting: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{(\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})} = \dfrac{\sqrt{a} - \sqrt{b}}{\sqrt{a}^2 - \sqrt{b}^2} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b} } We also note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a = b + 1} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a - b = 1} and the whole thing reduces to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{1978} - 1 }