# Arithmetic Mean

It two numbers a and b are in AP then their arithmetic mean (AM) =

In general, if a, b, c,â€¦n terms are in AP, then their AM =

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

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 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

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 abc,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 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 = r =
Hence, the required means are p , p , â€¦
p .
The above written means are nothing but prpr2,â€¦, prn, where r =

# Harmonic Mean

If two numbers a and b are in HP then their harmonic mean (HM) =

The process of inserting 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 =
1. AM, GM and HM will be in a GP.

So, GM is the geometric mean of this series.

And, GM =  [True only for two terms]
2. AM â‰¥ GM â‰¥ HM [Always true for any no. of terms]
3. The equation having a and b as its roots is x2 â€“ 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 =

Hence= 1 + 2 + 3 +â€¦.+n =
2. The sum of the squares of the 1st n natural numbers:

Sn =

Hence= 12 + 22 + 32+â€¦+ n2

3. 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

4. 22 + 42 + 62+â€¦+n terms

= 1/4