Maths-

General

Easy

Question

# If a_{1}, a_{2}, a_{3}, ........., a_{n} are positive real numbers whose product is a fixed number c, then the minimum value of a_{1} + a_{2} + a_{3} + .... + a_{n – 1} + 2a_{n} 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.

### Related Questions to study

maths-

### The domain of the function is

### The domain of the function is

maths-General

maths-

### Range of the function is

### Range of the function is

maths-General

maths-

### The domain of the function is

### The domain of the function is

maths-General

maths-

### Let , and , then dom is given by

### Let , and , then dom is given by

maths-General

physics-

### In the adjacent diagram, represents a wavefront and and , the corresponding two rays. Find the condition on for constructive interference at P between the ray and reflected ray

### In the adjacent diagram, represents a wavefront and and , the corresponding two rays. Find the condition on for constructive interference at P between the ray and reflected ray

physics-General

chemistry-

### Select the rate law that correspond to the data shown for the following reaction.

### Select the rate law that correspond to the data shown for the following reaction.

chemistry-General

chemistry-

### Select the rate law that corresponds to the data shown for the following reaction.

### Select the rate law that corresponds to the data shown for the following reaction.

chemistry-General

maths-

### is relation on given by Which of the following belongs to ?

### is relation on given by Which of the following belongs to ?

maths-General

physics-

### Two magnets A and B are identical and these are arranged as shown in the figure. Their length is negligible in comparison with the separation between them. A magnetic needle is placed between the magnets at point P which gets deflected through an angle under the influence of magnets. The ratio of distances d_{1} and d_{2} will be

### Two magnets A and B are identical and these are arranged as shown in the figure. Their length is negligible in comparison with the separation between them. A magnetic needle is placed between the magnets at point P which gets deflected through an angle under the influence of magnets. The ratio of distances d_{1} and d_{2} will be

physics-General

physics-

### Two small magnets X and Y of dipole moments M_{1} and M_{2} are fixed perpendicular to each other with their north poles in contact. This arrangement is placed on a floating body so as to move freeely in earth's magnetic field as shown in figure then the ratio of magnetic moment is

### Two small magnets X and Y of dipole moments M_{1} and M_{2} are fixed perpendicular to each other with their north poles in contact. This arrangement is placed on a floating body so as to move freeely in earth's magnetic field as shown in figure then the ratio of magnetic moment is

physics-General

physics-

### Find the resultant magnetic moment for the following arrangement

### Find the resultant magnetic moment for the following arrangement

physics-General

physics-

### A modulating signal is a square wave as shown in the figure

The carrier wave is given by The modulation index is

### A modulating signal is a square wave as shown in the figure

The carrier wave is given by The modulation index is

physics-General

physics-

### In the P-V diagram shown in figure ABC is a semicircle. The work done in the process ABC is

### In the P-V diagram shown in figure ABC is a semicircle. The work done in the process ABC is

physics-General

physics-

### Volume versus temperature graph of two moles of helium gas is as shown in figure. The ratio of heat absorbed and the work done by the gas in process 1-2 is

### Volume versus temperature graph of two moles of helium gas is as shown in figure. The ratio of heat absorbed and the work done by the gas in process 1-2 is

physics-General

physics-

### P-V diagram of an ideal gas is as shown in figure. Work done by the gas in process ABCD is

### P-V diagram of an ideal gas is as shown in figure. Work done by the gas in process ABCD is

physics-General