# Arithmetic Mean

It two numbers

*a*and*b*are in AP then their arithmetic mean (AM) =*a, b, c*,â€¦

*n*terms are in AP, then their AM =

*n*terms is nothing but the average of

*n*terms.

Hence, the sum of

*n*AMs between P and Q=

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=

Hence the sum =

Inserting a given number of arithmetic means between two given quantities
Let
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.*d*be the common difference;then

*q*= the (*n*+ 2)th term =*p*+ (*n*+ 1)*d*;Hence,

*d*=And the inserted means are:

*p*+ ,

*p*+ 2 ,â€¦,

*p*+

*n*.

# Geometric Mean

If two numbers

*a*and*b*are in GP, then their geometric mean (GM) =.In general, if

*a*,*b*,*c*,â€¦*n*terms are in GP, then their GM =.So, if three terms

*a, b*and*c*are in GP, then their GM =*b*=.Inserting a given number of geometric means between two given quantities

Let

Let us assume that r is the common ratio.

*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*=

*pr*

^{n+}^{1};

So,

*r*^{n}^{+1}= ;*r*=Hence, the required means are

*p**, p**, â€¦**p*.

The above written means are nothing but

*pr*,*pr*^{2},â€¦,*pr**, where*^{n}*r*=

# Harmonic Mean

If two numbers

*a*and*b*are in HP then their harmonic mean (HM) =*n*HMs between two given numbers is quite similar to the process of inserting AMâ€™s between two given numbers.

Relationship among AM, GM and HM

Now we know that for any two given numbers

*a*and*b*,AM =

GM =

HM =

- AM, GM and HM will be in a GP.
- AM â‰¥ GM â‰¥ HM [Always true for any no. of terms]
- The equation having
*a*and*b*as its roots is*x*^{2}â€“ 2A*x*+ G^{2}= 0.

Sum of n terms of some special series

In this part of progression, we shall see the sum of some other special sequences.

- The sum of the 1st
*n*natural numbers:*S*=_{n}*n*= - The sum of the squares of the 1st
*n*natural numbers:*S*=_{n}^{2}+ 2^{2}+ 3^{2}+â€¦+*n*^{2}^{= } ^{The sum of the cubes of the 1st n natural numbers:}^{Sn = (Sum of the 1st n natural numbers)2}^{= }^{It can be seen that (1 + 2 + 3)2 = 13 + 23 + 33}^{Hence, = 13 + 23 + 33 +â€¦+ n3 }- 2
^{2}+ 4^{2}+ 6^{2}+â€¦+*n*terms