FMM18

Friday Morning Math Problem

Radical Sums

Evaluate the sum:

${\dfrac {1}{1+{\sqrt {2}}}}+{\dfrac {1}{{\sqrt {2}}+{\sqrt {3}}}}+{\dfrac {1}{{\sqrt {3}}+{\sqrt {4}}}}+\dots +{\dfrac {1}{{\sqrt {1977}}+{\sqrt {1978}}}}$

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: ${\dfrac {1}{{\sqrt {a}}+{\sqrt {b}}}}={\dfrac {({\sqrt {a}}-{\sqrt {b}})}{({\sqrt {a}}+{\sqrt {b}})({\sqrt {a}}-{\sqrt {b}})}}={\dfrac {{\sqrt {a}}-{\sqrt {b}}}{{\sqrt {a}}^{2}-{\sqrt {b}}^{2}}}={\dfrac {{\sqrt {a}}-{\sqrt {b}}}{a-b}}$ We also note that $a=b+1$, so $a-b=1$ and the whole thing reduces to ${\sqrt {1978}}-1$