Mathematics ARITHMETIC , GEOMETRIC AND HARMONIC MEANS BETWEEN TWO GIVEN NUMBERS

Arithmetic Mean (`A.M.`) and insert `'n'` `AM's` between `a` and `b` :

`text(Definition :)`

When three quantities are in `A.P.` then the middle one is called the Arithmetic Mean of the other two.

e.g. `a, b, c` are in `A.P.` then `'b'` is the arithmetic mean between `'a'` and `'c' ` and `a+ c = 2b`.

It is to be noted that between two given quantities it is always possible to insert any number of terms such that the whole series thus formed shall be in `A.P.` and the terms thus inserted are called the `text(arithmetic means.)`

`text(To insert 'n' AM's between a and b :)`

Let `A_1, A_2, A_3 ........ A_n` are then means between `a` and `b.` Hence `a,A_1A_2, ........ A_n b` is an `A.P.` and `b` is the `(n + 2)^(th)` terms.

Hence `b=a+(n+1)d=>d=(b-a)/(n+1)`

Now `A_1=a+d`

`A_2=a+2d`

`vdots`

`A_n=a+nd`

`=> sum_(i=1)^nA_i=na+(1+2+3+....+n)d`

`=na+(n(n+1))/2d=na+(n(n+1))/2 . (b-a)/(n+1)`

`=n/2[2a+b-a]=n((a+b)/2)=nA`

Hence the sum of `nquad AM's` inserted between `a` and `b` is equal to `n` times a single `AM` between them.

Geometric Mean (`G.M.`) and insert `'n'` `GM's` between `a` and `b` :

`text(Definition :)`

If `a, b, c` are three positive numbers in `G.P.` then `b` is called the geometrical mean between `a` and `c` and `b^2= ac` . If `a` and `b` are two positive real numbers and `G` is the `G.M.` between them, then

`G^2 = ab`

`text(To insert 'n' GM's between a and b :)`

Let `a` and `b` are two positive numbers are `G_1, G_2, ... G_n` are `'n' GM's` then

`a , G_1 , G_2 ........... G_n, b` is a `G.P.` with `'b'` as its `(n+2)^(th)` term.

Hence `b=ar^(n+1)`

`:. r=(b/a)^(1/(n+1))`

Now `G_1=ar, G_2=ar^2,.......G_n=ar^n`

Hence `Pi_(i=1)^n G_i=a^n` `r^(1+2+.........+n)=a^n . r^(n(n+1)/2)=a^n[(b/a)^(1/(n+1))]^((n(n+1))/2)`

`=a^n . (b^(n//2))/(a^(n//2))=a^(n//2)b^(n//2)=(sqrt(ab))^n=G^n`

where `G` is the angle `GM` between `a` and `b`.

Hence product of `n quadGM's` inserted between of `a` and `b` is equal to the `n^(th)` power of a single `GM` between them.

Harmonic Mean (`H .M.`) and insert `'n'` `HM's` between `a` and `b` :

`text(Definition :)`

If `a, b, c` are in `H.P.` then middle term is called the harmonic mean between them. Hence if `H` is the harmonic mean (`H .M.`) between `a` and `b` then `a, H, b` are in `H.P.` and `H =(2ab)/(a+b)`.

(Recall that `AM = (a+b)/2` and `GM = sqrt(ab)` if `a > 0, b > 0`)

`text(To insert 'n' HM's between a and b :)`

Let `H_1, H_2 ........ H_n` are `n` `HM's` between `a` and `b`

hence `a, H_1, H_2, ..... H_n, b` are in `H.P.`

`1/a,1/H_1,1/H_2...................... 1/H_n,1/b` are in `A.P.`

`1/b=1/a+(n+1)d; -1/a=(n+1)d; d=(a-b)/(ab(n+1))`

`1/H_1=1/a+d`

`1/H_2=1/a+2d`

`1/H_3=1/a+3d`

`vdots`

`1/H_n=1/a+nd`

`sum_(i=1)^n1/H_i=n/a+(d(n)(n+1))/2=n/a s+(n(n+1))/2 . ((a-b))/(ab(n+1))`

`=n[1/a+(a-b)/(2ab)]=n/(2ab)[2b+a-b]=(n(a+b))/(2ab)=n . 1/H`

Hence sum of the reciprocals of all then `HM's` between `a` and `b` is equal to `n` times the reciprocal of single `HM` between `a` and `b`.