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 a_{1}, a_{2}, ... , a_{n} is ; A = .
n^{ }^{ }Arithmetic Means Between Two Numbers :
If a, b are any two given numbers & a, A_{1}, A_{2}, .... , A_{n}, ^{ }b ^{ }are in AP then A_{1}, A_{2}, ... A_{n} are the n ^{ }AM's ^{ }between ^{ }a & b .
A_{1} = a +^{} ,A_{2} = a +^{} , ...... , A_{n} = a +^{}
= a + d , = a + 2^{ }d , ...... , A_{n} = 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.
A_{r} = 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 .
nGeometric Means Between a , b : If a, b are two given numbers & a, G_{1}, G_{2}, .....^{ }, G_{n}, b are in GP . Then G_{1}, G_{2}, G_{3 }, ...., G_{n} are n^{ } GMs between a & b.
G_{1} = a(b/a)^{1/n+1} , G_{2} = a(b/a)^{2/n+1}, ...... , G_{n} = a(b/a)^{n/n+1}
= ar ,= ar² , ...... ^{ } = ar^{n}, 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 ⇒ 2n^{2}  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 = 16t^{2} – 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 22^{nd} 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
∴
⇒ H_{1} = ;
⇒ H_{2} =
⇒ H_{3} = ;
⇒ H_{4} = .
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1. What is the definition of arithmetic mean? 
2. How is the arithmetic mean calculated? 
3. What are the applications of arithmetic mean in real life? 
4. What are the limitations of the arithmetic mean? 
5. How is the arithmetic mean different from the median and mode? 

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