Doc: Arithmetic Mean

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Arithmetic, Geometric & Harmonic Means (AM, GM & HM)

Arithmetic Mean : If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP,  b is AM of a & c.

AM for any n positive number a1, a2, ... , an is ; A =  .

n - Arithmetic Means Between Two Numbers :

If a, b are any two given numbers & a, A1, A2, .... , An,  b  are in AP then A1, A2, ... An are the n  AM's  between  a & b .

A1 = a + ,A2 = a + , ...... , An = a +

= a + d , = a + 2 d  , ...... , An = a + nd , where d =

Note : Sum of n AM's inserted between a & b  is equal to n times the single AM between a & b i.e.

Ar = nA where A is the single AM between  a & b .

Geometric Means : If a, b, c are in GP, b  is the GM between a & c .

n-Geometric Means Between a , b : If a, b are two given numbers & a, G1, G2, ..... , Gn, b are in GP . Then G1, G2, G, ...., Gn are n  GMs between a & b.

G1 = a(b/a)1/n+1  , G2 = a(b/a)2/n+1, ...... , Gn = a(b/a)n/n+1

= ar ,= ar² , ......   = arn, where r = (b/a)1/n+1

Note : The product of n GMs between a & b is equal to the nth power of the single GM between a

Harmonic Mean : If a, b, c are in HP, b  is the HM between  a & c, then b = 2ac/[a + c].

Ex.30 Two consecutive numbers from 1, 2, 3,...., n are removed. The arithmetic mean of the remaining numbers is . Find n and those removed numbers.

Sol. Let p and (p + 1) be the removed numbers from 1, 2,...,n then sum of the remaining numbers

=  - (2p + 1)

From given condition      ⇒ 2n2 - 103n - 8p + 206 = 0

Since n and p are integers so n must be even let n = 2r. we get p =

Since p is an integer then (1 - r) must be divisible by 4. Let r = 1 + 4t, we get

n = 2 + 8t and p = 16t2 – 95t + 1,

Now 1 ≤ p < n ⇒ 1 ≤ 16t2 – 95t + 1 < 8t + 2

⇒ t = 6 ⇒ n= 50 and p =7

Hence removed numbers are 7 and 8.

Ex.31 Between two numbers whose sum is , an even number of A.M.'s is inserted, the sum of these means exceeds their number by unity. Find the number of means.

Sol. Let a and b be two numbers and 2n A.M.'s are inserted between a and b then

(a + b) = 2n + 1⇒ n  = 2n + 1.      ⇒ n = 6

Number of means = 12.

Ex.32 Insert 20 A.M. between 2 and 86.

Sol. Here 2 is the first term and 86 is the 22nd term of A.P. so 86 = 2 + (21) d ⇒ d = 4

so the series is 2, 6, 10, 14,......., 82, 86q  required means are 6, 10, 14,.....82

Ex.33 A, B and C are distinct positive integers, less than or equal to 10. The arithmetic mean of A and B is 9. The geometric mean of A and C is . Find the harmonic mean of B and C.

Sol. A + B = 18 .....(1)

AC = 72 .....(2)

There are only two possibilities A = 10 and B = 8 or A = 8 and B = 10

If A = 10 then from (2) C is not an integer.

Hence A = 8 and B = 10;  C = 9

∴ H.M. between B and C =  =

Ex.34 Insert 4 H.M. between .

Sol. Let d be the common difference of corresponding A.P.  so d =  = 1

⇒ H1 =  ;

⇒  H2 =

⇒   H3 =  ;

⇒   H4 = .

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## Mathematics (Maths) Class 11

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## FAQs on Doc: Arithmetic Mean - Mathematics (Maths) Class 11 - Commerce

 1. What is the definition of arithmetic mean?
Ans. The arithmetic mean, also known as the average, is a statistical measure that represents the sum of a set of numbers divided by the count of those numbers. It is commonly used to find the central tendency or average value of a dataset.
 2. How is the arithmetic mean calculated?
Ans. To calculate the arithmetic mean, you need to sum up all the numbers in the dataset and then divide the sum by the count of numbers in the dataset. For example, if you have the numbers 2, 4, 6, and 8, you would add them up (2+4+6+8=20) and then divide by 4 (total count) to get an arithmetic mean of 5.
 3. What are the applications of arithmetic mean in real life?
Ans. The arithmetic mean has various applications in real life. It is commonly used in financial analysis to calculate average stock prices, in sports to determine the average score or performance of players, and in academic grading to determine the average marks of students. Additionally, it is used in market research to analyze consumer preferences and in quality control to monitor production processes.
 4. What are the limitations of the arithmetic mean?
Ans. While the arithmetic mean is a useful measure, it has certain limitations. One limitation is that it can be influenced by extreme values or outliers in the dataset, causing it to be skewed. Another limitation is that it may not accurately represent the dataset if it contains values with different magnitudes or scales. Additionally, the arithmetic mean does not provide information about the dispersion or variability of the data.
 5. How is the arithmetic mean different from the median and mode?
Ans. The arithmetic mean, median, and mode are all measures of central tendency, but they represent different aspects of the data. The arithmetic mean is calculated by summing up all the values and dividing by the count, while the median is the middle value when the data is arranged in ascending or descending order. The mode, on the other hand, is the value that appears most frequently in the dataset. The arithmetic mean is sensitive to extreme values, the median is less affected by outliers, and the mode is useful for categorical data or when finding the most common value.

## Mathematics (Maths) Class 11

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