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If a(i) gt 0 (i = 1,2,3,….n) prove that ...

If `a_(i) gt 0` `(i = 1,2,3,….n)` prove that
`sum_(1 le i le j le n) sqrt(a_(i)a_(j)) le (n - 1)/(2) (a_(1) + a_(2) + …. + a_(n))`

Text Solution

Verified by Experts

Using `A.M. ge G.M`., we have
`2^(sin x)+2^(cosx) ge 2sqrt(2^sinx 2^cos x)=2sqrt(2^(sinx+cosx))`
Now we know that
`sin x +cos x ge -sqrt(2)`
`rArr 2^(sinx)+2^(cos x) ge 2sqrt(2^-sqrt(2))`
Hence, the minimum value of `2^sinx +2^cosx is 2(1(1)/(sqrt(2)))`
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