Progression (Sequence and Series)Medium
Question
If a1, a2 ..... an are postive real number whose product is a fixed number c, them the minimum value of a1 + a2 + ...... + an-1 +2an is
Options
A.n(2c)1/n
B.(n + 1)c1/n
C.2nc1/n
D.(n + 1)(2c)1/n
Solution
Given, a1, a2 ..... an = c
⇒ a1a2a3 ...... (an - 1)(2an) = 2c .....(i)
∴
(a1.a2.a3 ...... 2an)1/n (using AM ≥ GM)
∴ a1 + a2 + a3 + ...... + 2an ≥ n(2c)1/n [from eq. (i)]
⇒ minimum value of
a1 + a2 + a3 + ...... + 2an = n(2c)1/n
⇒ a1a2a3 ...... (an - 1)(2an) = 2c .....(i)
∴
(a1.a2.a3 ...... 2an)1/n (using AM ≥ GM)∴ a1 + a2 + a3 + ...... + 2an ≥ n(2c)1/n [from eq. (i)]
⇒ minimum value of
a1 + a2 + a3 + ...... + 2an = n(2c)1/n
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