Progression (Sequence and Series)Medium
Question
The product of n positive numbers is unity, then their sum is
Options
A.a positive integer
B.divisible by n
C.equal to n + 
D.never less than n
Solution
Since, product of n positive numbers is unity.
⇒ x1. x2. x3 ...... xn = 1 ...... (i)
Using AM ≥ GM
⇒
1(x1. x2 ......xn)1/n
⇒ x1 + x2 + ......+ xn ≥ n(1)1/n [from Eq. (i)]
⇒ Sum of n positive number is never less than n.
⇒ x1. x2. x3 ...... xn = 1 ...... (i)
Using AM ≥ GM
⇒
1(x1. x2 ......xn)1/n ⇒ x1 + x2 + ......+ xn ≥ n(1)1/n [from Eq. (i)]
⇒ Sum of n positive number is never less than n.
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