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If a(i) gt 0 AA I in N such that prod(i=...

If `a_(i) gt 0 AA I in N` such that `prod_(i=1)^(n) a_(i) = 1`, then prove that `(a + a_(1)) (1 + a_(2)) (1 + a_(3)) .... (1 + a_(n)) ge 2^(n)`

Text Solution

Verified by Experts

Using A.M. `ge G.M.`
`1 + a_(1) ge 2 sqrt(a_(1))`
`1 + a_(2)ge2sqrt(a_(2))`
`vdots`
`1 + a_(n) ge 2 sqrt(a_(n)) rArr (1 + a_(1)) (1 + a_(2)) (1 + a_(n)) ge 2^(n) (a_(1) a_(2) a_(3) ... A_(n))^(1//2)`
As `a_(1) a_(2) a_(3).... a_(n) = 1`
Hence `(1 + a_(1)) (1 + a_(2)).... (1 + a_(n)) ge 2^(n)`
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