Answer
Verified
430.5k+ views
Hint: Here, we will first find the arithmetic mean and geometric mean of the terms in the expression. Then, we will use the relation between arithmetic mean and geometric mean to form an inequation. Finally, we will use the given information to find the minimum value of the expression \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\].
Formula Used:
We will use the following formulas:
The arithmetic mean of the \[n\] numbers \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] is given by the formula \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + {a_n}}}{n}\].
The geometric mean of the \[n\] numbers \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] is given by the formula \[G.M. = \sqrt[n]{{{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}}}\].
Complete step-by-step answer:
We will use the formula for A.M. and G.M. to find the minimum value of \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\].
The number of terms in the sum \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n\].
Therefore, using the formula \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + {a_n}}}{n}\], we get the arithmetic mean as
\[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}\]
The number of terms in the sum \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n\].
Therefore, using the formula \[G.M. = \sqrt[n]{{{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}}}\], we get the geometric mean as
\[G.M. = {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\]
Now, we know that the arithmetic mean is always greater than or equal to the geometric mean.
Therefore, we get
\[ \Rightarrow A.M. \ge G.M.\]
Substituting \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}\] and \[G.M. = {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\] in the inequation, we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\]
Rewriting the inequation, we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}} \right)^{1/n}}\]
It is given that the number \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] are positive real numbers whose product is a fixed number \[c\].
Therefore, we get
\[{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n} = c\]
Substituting \[{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n} = c\] in the inequation \[\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}} \right)^{1/n}}\], we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2c} \right)^{1/n}}\]
Multiplying both sides by \[n\], we get
\[ \Rightarrow n\left( {\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}} \right) \ge n{\left( {2c} \right)^{1/n}}\]
Thus, we get
\[ \Rightarrow {a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n} \ge n{\left( {2c} \right)^{1/n}}\]
Therefore, the value of the expression \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is greater than or equal to \[n{\left( {2c} \right)^{1/n}}\].
Thus, the minimum value of the expression \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n{\left( {2c} \right)^{1/n}}\].
The correct option is option (a).
Note: We multiplied both sides of the inequation \[\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2c} \right)^{1/n}}\] by \[n\]. Since the number of terms cannot be negative, \[n\] is a positive integer. Therefore, we could multiply both sides of the inequation by \[n\] without changing the sign of the inequation.
Here we used geometric mean and arithmetic mean to solve the question. These are the two types of mean and the third type of mean is harmonic mean.
Formula Used:
We will use the following formulas:
The arithmetic mean of the \[n\] numbers \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] is given by the formula \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + {a_n}}}{n}\].
The geometric mean of the \[n\] numbers \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] is given by the formula \[G.M. = \sqrt[n]{{{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}}}\].
Complete step-by-step answer:
We will use the formula for A.M. and G.M. to find the minimum value of \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\].
The number of terms in the sum \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n\].
Therefore, using the formula \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + {a_n}}}{n}\], we get the arithmetic mean as
\[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}\]
The number of terms in the sum \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n\].
Therefore, using the formula \[G.M. = \sqrt[n]{{{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}}}\], we get the geometric mean as
\[G.M. = {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\]
Now, we know that the arithmetic mean is always greater than or equal to the geometric mean.
Therefore, we get
\[ \Rightarrow A.M. \ge G.M.\]
Substituting \[A.M. = \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}\] and \[G.M. = {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\] in the inequation, we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {{a_1}{a_2} \ldots \ldots {a_{n - 1}}2{a_n}} \right)^{1/n}}\]
Rewriting the inequation, we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}} \right)^{1/n}}\]
It is given that the number \[{a_1},{a_2}, \ldots \ldots ,{a_n}\] are positive real numbers whose product is a fixed number \[c\].
Therefore, we get
\[{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n} = c\]
Substituting \[{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n} = c\] in the inequation \[\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2{a_1}{a_2} \ldots \ldots {a_{n - 1}}{a_n}} \right)^{1/n}}\], we get
\[ \Rightarrow \dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2c} \right)^{1/n}}\]
Multiplying both sides by \[n\], we get
\[ \Rightarrow n\left( {\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n}} \right) \ge n{\left( {2c} \right)^{1/n}}\]
Thus, we get
\[ \Rightarrow {a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n} \ge n{\left( {2c} \right)^{1/n}}\]
Therefore, the value of the expression \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is greater than or equal to \[n{\left( {2c} \right)^{1/n}}\].
Thus, the minimum value of the expression \[{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}\] is \[n{\left( {2c} \right)^{1/n}}\].
The correct option is option (a).
Note: We multiplied both sides of the inequation \[\dfrac{{{a_1} + {a_2} + \ldots \ldots + {a_{n - 1}} + 2{a_n}}}{n} \ge {\left( {2c} \right)^{1/n}}\] by \[n\]. Since the number of terms cannot be negative, \[n\] is a positive integer. Therefore, we could multiply both sides of the inequation by \[n\] without changing the sign of the inequation.
Here we used geometric mean and arithmetic mean to solve the question. These are the two types of mean and the third type of mean is harmonic mean.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE