Question:
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}+2a_n$ is
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}+2a_n$ is
Updated On: Jun 14, 2022
- $ n\, (2c)^{1/n} $
- $ (n\, +1) c^{1/n} $
- $ 2cn^{1/n}$
- $ (n\, +1) (2c)^{1/n} $
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Given, $ a_1, a_2,...,a_n = c $
$\Rightarrow \, \, \, a_1\, a_2\, a_3...(a_{n-1})(2a_n) = 2c\hspace40mm ...(i)$
$\therefore \, \, \, \, \frac{a_1\, + \, a_2\, +\, a_3\, +...+2a_n}{n} \ge (a_1.a_2.a_3...2a_n)^{1/n}$
$ \hspace70mm [using\, AM \ge GM] $
$ \Rightarrow \, \, \, a_1+ a_2+ a_3+...+2a_n \ge \, n(2c)^{1/n} \hspace5mm [from\, E (i)] $
$ \Rightarrow \, \, \, $ Minimum value of
$ \hspace30mm a_1+ a_2+ a_3+...+2a_n = \, n(2c)^{1/n}$
$\Rightarrow \, \, \, a_1\, a_2\, a_3...(a_{n-1})(2a_n) = 2c\hspace40mm ...(i)$
$\therefore \, \, \, \, \frac{a_1\, + \, a_2\, +\, a_3\, +...+2a_n}{n} \ge (a_1.a_2.a_3...2a_n)^{1/n}$
$ \hspace70mm [using\, AM \ge GM] $
$ \Rightarrow \, \, \, a_1+ a_2+ a_3+...+2a_n \ge \, n(2c)^{1/n} \hspace5mm [from\, E (i)] $
$ \Rightarrow \, \, \, $ Minimum value of
$ \hspace30mm a_1+ a_2+ a_3+...+2a_n = \, n(2c)^{1/n}$
Was this answer helpful?
0
0
Top Questions on Series
- The value of \[ \lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k + 3)!} \] is:
- In the expansion of $(1 + x)^n$, $\frac{{}^nC_1}{{}^nC_0} + 2 \cdot \frac{{}^nC_2}{{}^nC_1} + 3 \cdot \frac{{}^nC_3}{{}^nC_2} + \dots + n \cdot \frac{{}^nC_n}{{}^nC_{n-1}}$ is equal to:
- If \( \sum_{k=1}^{n} k(k+1)(k-1) = pn^4 + qn^3 + tn^2 + sn \), where \( p, q, t, s \) are constants, then the value of \( s \) is equal to:
- If \( A = 1 + r^a + r^{2a} + r^{3a} + \dots \infty \) and \( B = 1 + r^b + r^{2b} + r^{3b} + \dots \infty \), then \( \frac{a}{b} \) is equal.
- The sum of the infinite series \(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \dots\) is equal to:
View More Questions
Questions Asked in JEE Advanced exam
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
View More Questions
Concepts Used:
Series
A collection of numbers that is presented as the sum of the numbers in a stated order is called a series. As an outcome, every two numbers in a series are separated by the addition (+) sign. The order of the elements in the series really doesn't matters. If a series demonstrates a finite sequence, it is said to be finite, and if it demonstrates an endless sequence, it is said to be infinite.
Read More: Sequence and Series
Types of Series:
The following are the two main types of series are: