Maths-
General
Easy
Question
If a1, a2, a3, ........., an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + a3 + .... + an – 1 + 2an is
Hint:
In this question, we have to find the minimum value of a1 + a2 + a3 + …… + an-1 + 2an. For solving this question, remember the inequality of A.M and G. M.
The correct answer is: 
Here we have to find the minimum value of a1 + a2 + a3 + …… + an-1 + 2an.
Firstly, given is
a1. a2. a3….an =c ------(1)
We know that,
Always A.M ≥ G.M
So, A.M =
And
G.M = (a1.a2.a3……an-1 (2an)) ^
From, A.M ≥ G.M, we can write
=> (a1 + a2 + a3 + …+ an-1 + 2an) / n ≥ (a1.a2.a3……an-1 (2an)) ^
Cross multiply,
=> a1 + a2 + a3 + …+ an-1 + 2an ≥ n (a1.a2.a3……an-1 (2an)) ^
=> a1 + a2 + a3 + …+ an-1 + 2an ≥ n 2 (c)^
[since, c = a1. a2. a3…...an]
=> a1 + a2 + a3 + …+ an-1 + 2an ≥ 2nc^
Therefore, the minimum value of a1 + a2 + a3 + …… + an-1 + 2an is 2nc^
.
The correct answer is 2nc^
.
In this question, we have to find that minimum value of a1 + a2 + a3 + …… + an-1 + 2an. Remember always the inequality of A.M and G.M. Always A.M ≥ G.M.
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