Question 25: Compare \(\sqrt 1 + \sqrt 2 + \sqrt 3 + … + \sqrt {25} \) and 75.

We have:

\(\sqrt 1 + \sqrt 2 + \sqrt 3 > \sqrt 1 + \sqrt 1 + \sqrt 1\)\( = 1 + 1 + 1 = 3\)

\(\sqrt 4 + \sqrt 5 + … + \sqrt 8 > \)\(\underbrace {\sqrt 4 + \sqrt 4 + … + \sqrt 4 }_{5\,number\,rank }\)\( = \underbrace {2 + 2 + … + 2}_{5\,number\,rank} = 10\)

\(\sqrt 9 + \sqrt {10} + … + \sqrt {15} >\)\( \underbrace {\sqrt 9 + \sqrt 9 + … + \sqrt 9 }_{7\, number\,rank}\)\( = \underbrace {3 + 3 + … + 3}_{7\,number\,rank} = 21\)

\(\sqrt {16} + \sqrt {17} + … + \sqrt {24} >\)\( \underbrace {\sqrt {16} + \sqrt {16} + … + \sqrt { 16} }_{9\,number\,rank}\)\( = \underbrace {4 + 4 + … + 4}_{9\,number\,rank} = 36\) and \(\sqrt {25} = 5\)

Hence: \(\sqrt 1 + \sqrt 2 + \sqrt 3 + … + \sqrt {25} >\)\( 3 + 10 + 21 + 36 + 5 = 75.\)