Type of Means

“Mathematics is the queen of science and arithmetic is the queen of mathematics.”  These were the words of Carl Gauss, the person who gave the world sequence and series arithmetic progression concept along with several other major contributions in the field of mathematics.), A.P is a special type of sequence where the difference between the two consecutive terms is always the same/Constant. In arithmetic progression the next term is obtained by adding or subtracting the common difference to or from the previous term whereas when we are multiplying OR dividing the previous term with any non-zero constant which is called as the common ratio then the sequence thus obtained is said to be Geometric progression (G.P.). After working with A.P and G.P one progresses towards arithmetic mean, geometric mean and eventually to harmonic mean. Let us see one by one the concepts 

Arithmetic progression

A.P is a special type of sequence where the difference between the two consecutive terms is always the same/Constant. And this difference is said to be a common difference and is denoted by d. 

Consider this below sequence which is in A.P

1, 2, 3, 4, 5 ………n

Here the first term is 1, which we will denote as t1

t2 = second term = 2 

t5 = fifth term = 5

tn = nth term

to find the common difference we do t2 – t1 or we can say we would do tn – tn-1

i.e., 2 -1 =1 is the common difference(d) here

Let’s try to formulate the formula for A.P using our current understanding of our topic.

If t1 is denoted as a (first term) and common difference as d then 

a+d = t2 

t3 = t2 + d = a + d +d = a +2d 

t4 = t3 + d = a +3d

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tn = tn-1 + d = a + (n-1)d

and the sum of AP is given by SN = N2[ 2a+N-1d]

Geometric progression

In arithmetic progression the next term is obtained by adding or subtracting the common difference to or from the previous term whereas when we are multiplying OR dividing the previous term with any non-zero constant which is called as the common ratio then the sequence thus obtained is said to be Geometric progression (G.P.).

Consider the following series 

1, 2, 4, 8, 16, 32 …….

Here, first term = a = 1 

Now If we carefully observe 1*2 = 2 = second term (a2)

2 *2 = 4 = third term (a3) 

4*2 = 8 = fourth term (a4)

8*2= 16 = fifth term and so on 

The constant term 2 that we are multiplying with each term here in this series is called common ratio and is denoted by r. 

Basically, if we write it in terms of a and r our series would look something like below

a, ar, ar2, ar3, ar4 and so on till arn-1 

Harmonic progression (H.P.)

A sequence let’s say t1 , t2 , t3 , ………… , tn will be called harmonic progression if 

1t1, 1t2 , 1t3………1tn are in A.P (ARITHMETIC PROGRESSION) 

Example: 14, 314,316 are in H.P as 4, 143 , 163 are in A.P.

1. Find the nth term of H.P 15, 1 , -13 , -17

Solutions: as we can see 5 , 1 , -3, -7 are in A.P

With a = 5 and d = -4

Hence tn = a + (n -1)d = 5 + (n -1) (-4) = 5 -4n +4 = 9 – 4n

Therefore , nth term of H.P = 19-4n

Types of mean 

• Arithmetic mean or A.M
• Geometric mean or G.M.
• Harmonic mean or H.M.

Arithmetic mean or A.M.

If x and y are two numbers their Arithmetic mean or A.M. is given by A = X+Y2

It can be clearly seen, x , A , y form an A.P.

That is , A-X = Y – A 

2A = X +Y 

A = X+Y2

For n numbers 

A = x1+x2+x3………+xnn

Geometric mean or G.M.

If x and y are two numbers which have the same sign then their G.M. is given by G = xy

Here also x , G,y form a G.P

That is, Gx=yG 

G2 = x y 

Or 

by G = xy

Harmonic mean or H.M.

If x and y are two numbers then their H.M. is given as 

H = 2xyx+y

We observe that x, H, y forms an HP.

That is 1x , 1H , 1y are in A.P

After solving the above A.P. we get , H = 2xyx+y

If x = y then A = G = H 

Relationship between AM and GM

Let x and y be two numbers then, 

AM = x+y2  and GM =√ xy

AM – GM  = x+y2 – √ xy =

x+y-2√xy/2 = x-y22 

and this is always greater than 0 

Therefore AM is always equal to or greater than GM

EXAMPLES:

1. Find the A.M, G.M , H.M Of the numbers 4 and 16

Ans: here x = 4 and y = 16

A = X+Y/2= 4+16/ 2 = 20/2 = 10

G = √xy = 8

H = 2xy/x+y = 128/20 = 32/5

1. Insert 4 terms between 2 and 22 so that the new sequence is in AP?

Ans: let a1 ,a2,a3,a4 be the four terms between 2 and 22 and we know that this whole sequence is in AP

So, first term = a = 2 , last term = l = 22 , n = 6

22= 2 + (6-1) d= 20= 5d 

D = 4

a = 2 , a1 = 2+4 = 6

a2 = 6+4 = 10 , a3= 10+4 = 14

a4= 14+4 = 18 , a5 = 18+4 = 22

Therefore, the terms between 2 and 22 are 6, 10, 14, 18.

Conclusion

A sequence of numbers is called a Geometric progression if the ratio of any two consecutive terms is always constant. The behavior of a geometric sequence depends on the value of the common ratio. AM is always equal to or greater than GM. After working with A.P and G.P one progresses towards arithmetic mean, geometric mean and eventually to harmonic mean. If x and y are two numbers then their H.M. is given as 

H = 2xy/x+y