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Arithmetic Mean

It two numbers a and b are in AP then their arithmetic mean (AM) = Description: 2764.png
 
In general, if a, b, c,…n terms are in AP, then their AM = Description: 2774.png
 
Its imperative to mention here is that, in simple terms, AM of n terms is nothing but the average of n terms.
 
Hence, the sum of n AMs between P and Q
Description: 2783.png
 
Example-1
If a, b, c, d, e and f are the AMs between 2 and 12, then find the value of a + b + c + d + e + f?
Solution
The Sum of n AMs between P and Q
Description: 2792.png 
Hence the sum = Description: 2796.png 
 
 
Inserting a given number of arithmetic means between two given quantities
 
Let p and q be the given quantities and n be the number of means which is to be inserted. After inserting n means, the total number of terms including the extremes will be equal to n + 2. Now we have to find a series of n + 2 terms in AP, of which p is the first, and q is the last term.
 
Let d be the common difference;
then q = the (n + 2)th term = p + (n + 1)d;
Hence, d Description: 2805.png
 
And the inserted means are:
p + Description: 2814.pngp + 2 Description: 2818.png,…, p + n Description: 2822.png.

Geometric Mean

If two numbers a and b are in GP, then their geometric mean (GM) =Description: 2826.png.
 
In general, if abc,…n terms are in GP, then their GM =Description: 2835.png.
 
So, if three terms a, b and c are in GP, then their GM = b =Description: 2845.png.
 
Inserting a given number of geometric means between two given quantities
 
Let p and q be the given quantities and n be the required number of geometric means to be inserted between p and q. In all, there will be n + 2 terms, so we have to find a series of n + 2 terms in a GP of which p is the first term and q is the last term.

Let us assume that r is the common ratio.
Then q is the (n + 2)th term.
q = prn+1;
So, rn+1 = Description: 2854.pngr = Description: 2863.png
Hence, the required means are p Description: 2872.png, p Description: 2876.png, … 
p Description: 2885.png.
The above written means are nothing but prpr2,…, prn, where r = Description: 2894.png

Harmonic Mean

If two numbers a and b are in HP then their harmonic mean (HM) =Description: 2898.png
 
The process of inserting n HMs between two given numbers is quite similar to the process of inserting AM’s between two given numbers.
 
Relationship among AM, GM and HM
 
Now we know that for any two given numbers a and b,
AM = Description: 2907.png
GM = Description: 2916.png 
HM = Description: 2921.png 
  1. AM, GM and HM will be in a GP.
     
    So, GM is the geometric mean of this series.
     
    And, GM = Description: 2925.png [True only for two terms]
  2. AM  GM  HM [Always true for any no. of terms]
  3. The equation having a and b as its roots is x2 – 2Ax + G2 = 0.
Sum of n terms of some special series
 
In this part of progression, we shall see the sum of some other special sequences.
  1. The sum of the 1st n natural numbers:
     
    Sn = Description: 2934.png
     
    HenceDescription: 2943.png= 1 + 2 + 3 +….+n = Description: 2952.png
  2. The sum of the squares of the 1st n natural numbers:
     
    Sn = Description: 2956.png
     
    HenceDescription: 2965.png= 12 + 22 + 32+…+ n2
     
    Description: 2974.png
  3. The sum of the cubes of the 1st n natural numbers:
     
    Sn = (Sum of the 1st n natural numbers)2
     
    Description: 2983.png
     
    It can be seen that (1 + 2 + 3)2 = 13 + 23 + 33
     
    Hence, Description: 2992.png= 13 + 23 + 33 +…+ n3 
     
    Description: 3002.png
  4. 22 + 42 + 62+…+n terms
     
    = 1/4 Description: 3006.png




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