If a1- a2- - ann - 3 are all unequal positive real numbers- and E - 1 - a1 - a121 - a2 - a22-1 - an - an2a1- a2- - an then which of the following best describes E-

The correct option is C E > 3^n a _ 1 , a _ 2 , a _ 3 ,... a _ n all are unequal n gt;3 E=( 1+ a _ 1 + a _ 1 ^ 2 ) ( 1+ a _ 2 + a _ 2 ^ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Summation by Sigma Method

If a1, a2, ...

Question

If $a_{1}, a_{2}, . . . . . . ., a_{n} (n > 3)$ are all unequal positive real numbers, and
$E = \frac{(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) . . . . . . (1 + a_{n} + a_{n}^{2})}{a_{1}, a_{2}, . . . . . . ., a_{n}}$
then which of the following best describes E?

E \leq 2^{n}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

E \geq 3^{n}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

E > 3^{n}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

E > 2^{n}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $E > 3^{n}$
$a_{1}, a_{2}, a_{3}, . . . a_{n}$ all are unequal $n > 3 E = \frac{(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) . . . (1 + a_{n} + a_{n}^{n})}{a_{1} . a_{2} . a_{3} . . . . a_{n}} = (\frac{1}{a_{1}} + 1 + a_{1}) (1 + \frac{1}{a_{2}} + a_{2}) (1 + \frac{1}{a_{3}} + a_{3}) . . . . (1 + \frac{1}{a_{n}} + a_{n}) A . M . \geq G . M . \frac{\frac{1}{a_{1}} + a_{1} + 1}{3} \geq \sqrt[3]{\frac{1}{a_{1}} . a_{1} .1} \frac{1}{a_{1}} + a_{1} + 1 \geq 3$
As $a_{1}, a_{2}, a_{3}, . . . a_{n}$ are unequal if $\frac{1}{a_{1}} + a_{1} + 1 = 3$ ,then all others must be $> 3 E = (\frac{1}{a_{1}} + 1 + a_{1}) (1 + \frac{1}{a_{2}} + a_{2}) . . . . (1 + \frac{1}{a_{n}} + a_{n}) > 3.3.....3 (n t i m e s) E > 3^{n}$

Suggest Corrections

Similar questions

$a_{1}, a_{2}, \dots \dots . a_{n}$ are A.P. Where $a_{1}$ > 0 for all i, then $\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + . . . . + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}}$

Q. If

a_{1}, a_{2}, a_{3} . . . . a_{15} \in R^{+}

and

a_{1}, a_{2}, a_{3} . . . a_{n} = 1

, then minimum value of

(1 + a_{1} + a_{1}^{2}) (1 + a_{3} + a_{3}^{2}) . . . . (1 + a_{a} + a_{a}^{2})

is equal to:_

Q. If

a_{1} + a_{2}, . . ., a_{n}

are positive numbers such that

a_{1} \cdot a_{2} . . . a_{n} = 1,

then their sum is :

Q. If

a_{1}, a_{2}, . . a_{n}

are positive real numbers whose product is a fixed number

c

, then the minimum value of

a_{1} + a_{2} +, . . a_{n - 1} + 2 a_{n}

Q. If

a_{1}, a_{2}, . . . . . . ., a_{n}

are positive real numbers whose product is a fixed number c, then the minimum value of

a_{1} + a_{2} + . . . . . . . + a_{n - 1} + 2 a_{n}

Join BYJU'S Learning Program

If a1,a2,.......,an(n>3) are all unequal positive real numbers, and E=(1+a1+a21)(1+a2+a22)......(1+an+a2n)a1,a2,.......,an then which of the following best describes E?

If $a_{1}, a_{2}, . . . . . . ., a_{n} (n > 3)$ are all unequal positive real numbers, and
$E = \frac{(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) . . . . . . (1 + a_{n} + a_{n}^{2})}{a_{1}, a_{2}, . . . . . . ., a_{n}}$
then which of the following best describes E?