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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, ... , a_n is ; A =  .

n - Arithmetic Means Between Two Numbers :

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

A₁ = a + ,A₂ = 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 .

n-Geometric Means Between a , b : If a, b are two given numbers & a, G₁, G₂, ..... , G_n, b are in GP . Then G₁, G₂, G₃ , ...., G_n are n  GMs between a & b.

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

∴

⇒ H₁ =  ;

⇒  H₂ =

⇒   H₃ =  ;

⇒   H₄ = .