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 ->  Inequalities -> 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      Log On


   

Question 1025262: Prove that:
( +sqrt%28+1%5E2%2B2%5E2%2B3%5E2+%29+ + +sqrt%282%5E2%2B3%5E2%2B4%5E2%29+ + ... + +sqrt%28+n%5E2%2B+%28n%2B1%29%5E2%2B%28n%2B2%29%5E2+%29+ )/ +sqrt%28+3+%29+ > n(n+3)/2
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
For , (this is easy to verify algebraically).

Then



The right hand side of the inequality equals . Dividing both sides by gives the desired result.