If three terms are in AP, then the middle term is called the Arithmetic Mean (A.M.) of the other two numbers. So if a, b, and c are in A.P., then b is AM of a and c. If three terms are in G.P., then the middle term is called the Geometric Mean (G.M.) of the other two numbers. So if a, b, and c are in G.P., then b is GM of a and c. In real life, we use the application of AM-GM Inequality in statistics, and economics and examine some optimization problems.
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In this article, we will cover the application of AM-GM Inequality. This category falls under the broader category of sequence and series, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main Exam (from 2013 to 2023), a total of nine questions have been asked on this concept, including one in 2019, one in 2020, one in 2021, two in 2022, and three in 2023.
What is AM-GM Inequality?
Arithmetic mean - Geometric mean (AM-GM) Inequality is a fundamental result in algebra that provides a relationship between the arithmetic mean and the geometric mean of a set of non-negative real numbers. This inequality states that for any list of non-negative real numbers, the arithmetic mean (average) is at least as great as the geometric mean.
To prove this we will be using the fact that A.M. ≥ G.M
So,
Let A, G, and H be arithmetic, geometric, and harmonic means of two positive real numbers a and b.
Then,
Note that A = G when a = b
Now,
Note :
If are n positive real numbers, then
A = A.M. of =
G = G.M. of =
H = H.M. of =
In such cases also
And A=G=H, when
Example 1: If then the least value of [JEE MAINS 2023]
Solution
Let
Hence, the answer is 4
Example 2: Let a, b, c, and d be positive real numbers such that .If the maximum value of is , then the value of is [JEE MAINS 2023]
Solution
Let's assume Numbers –
We know A.M. G.M.
Hence, the answer is 90
Example 3: If the minimum value of , is , then the value of is equal to : [JEE MAINS 2022]
Solution
Hence, the answer is 128
Example 4: Let . If , then the least value of is [JEE MAINS 2022]
Solution
we know that
the least value of
Hence, the answer is 40
Example 5: The minimum value of is: [JEE MAINS 2020]
Solution
Hence, the answer is
The Arithmetic Mean (AM) and Geometric Mean (GM) inequalities are fundamental principles in mathematics, offering insights into the relationships between arithmetic averages and geometric averages of sets of numbers. It helps us to solve problems ranging from simpler to advanced mathematical solutions. Understanding and applying these inequalities enriches our ability to analyze and derive conclusions in mathematical reasoning and practical applications alike.
Let A, G, and H be arithmetic, geometric, and harmonic means of two positive real numbers a and b.
Then,
Let A, G, and H be arithmetic, geometric, and harmonic means of two positive real numbers a and b.
Then,
A ≥ G ≥ H
The relations between Arithmetic, Geometric, and Harmonic mean if a=b is A = G = H
If three terms are in AP, then the middle term is called the Arithmetic Mean (A.M.) of the other two numbers. So if a, b, and c are in A.P., then b is AM of a and c. whereas, If three terms are in G.P., then the middle term is called the Geometric Mean (G.M.) of the other two numbers. So if a, b, and c are in G.P., then b is GM of a and c.
If A, G, and H of 2 positive real numbers form a geometric progression, the relation between arithmetic, geometric, and harmonic mean is G2 = AH.
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