SOLUTION: Prove that:
( {{{ sqrt( 1^2+2^2+3^2 ) }}} + {{{ sqrt(2^2+3^2+4^2) }}} + ... + {{{ sqrt( n^2+ (n+1)^2+(n+2)^2 ) }}} )/ {{{ sqrt( 3 ) }}} > n(n+3)/2
Algebra.Com
Question 1025262: Prove that:
( + + ... + )/ > n(n+3)/2
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
For 
, ^2 + (a+2)^2} > \sqrt{3 (a+1)^2} = (a+1) \sqrt{3})
(this is easy to verify algebraically).
Then
^2 + (n+2)^2} > \sqrt{3}(2+3+\ldots + (n+1)))
The right hand side of the inequality equals (n+2)}{2} - 1 \right) = \sqrt{3} \left( \frac{n(n+3)}{2} \right))
. Dividing both sides by 
gives the desired result.
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