Question

If a1, a2, a3, ..., an are n positive real numbers, then prove that (1 + a1) (1 + a2) …
(1 + an) > 1 + a1 + a2 +…+ an for n ≥ 2.

Answer 1

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SANKALP4516

31.01.2019

Math

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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

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anjali962

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isyllus Ambitious

Answer:

n(c)^{1/n}

Step-by-step explanation:

Given: If a₁, a₂, a₃, a₄, a₅, ........ are positive real number whose product is a fixed number c

Therefore,

a_1\cdot a_2\cdot a_3............\cdot a_{n-1}\cdot a_n\cdot=c

To find: minimum of a_1+a_2+a_3............+a_{n-1}+a_n

As we know,

AM ≥ GM

where, AM is arithmetic mean and GM is geometric mean.

Therefore,

\dfrac{a_1+a_2+a_3............+a_{n-1}+a_n}{n}\geq (a_1\cdot a_2\cdot a_3............\cdot a_{n-1}\cdot a_n)^{1/n}

a_1+a_2+a_3............+a_{n-1}+a_n\geq n(c)^{1/n}

Hence, The minimum value of sum is n(c)^{1/n}