Question

Let `a_(1),a_(2)…,a_(n)` be a non-negative real numbers such that `a_(1)+a_(2)+…+a_(n)=m` and let `S=sum_(iltj) a_(i)a_(j)`, then
A. `Sle(m^(2))/(2)`
B. `Sgt(m^(2))/(2)`
C. `Slt(m)/(2)`
D. `Sgt(m^(2))/(2)`

Answer 1

Correct Answer - A
We have,
`m^(2)=(a_(1)+a_(2)+...+a_(n))^(2)`
`implies" "m^(2)=underset(i=1)overset(n)suma_(i)""^(2)+2underset(iltj)overset(n)suma_(i)a_(j)`
`implies" "m^(2)=underset(i=1)overset(n)suma_(i)""^(2)+2" S"`
`implies" "m^(2)-2S=underset(i=1)overset(n)suma_(i)""^(2)impliesm^(2)-2Sge0impliesSle(m^(2))/(2)`