Mathematics Mean

Arithmetic Mean

If three terms are in AP, then the middle term is called the Arithmetic
Mean (or shortly written as AM) between the other two, so if a, b, care in AP,
then b is the AM of a and c.

`(i}` `text(Single AM of n Positive Numbers)`

Let n positive numbers be `a_1 , a_2 ,......... ,a_n` and A be the AM of these numbers,
then

`A = (a_1 + a_2 + ..... + a_n)/n`

In particular Let a and b be two given numbers and A be the AM between
them, then a, A, bare in AP.

`therefore` `A = (a+b)/2`

`text(Note)`

`1.` AM of `2 a, 3b , 5 c` is `(2a + 3b +5c)/3`

`2.` AM of `(a_1, a_2, a_3 , ....... , a(n _ 1), 2a_n)/n`

`(ii)` `text(Insert n-Arithmetic Mean Between Two Numbers)`

Let a and b be two given numbers and `A_1,A_2,A_3 , ... ,A_n` are AM's between
them.

Then, `a, A_1 , A_2 , A_3 , ... , A_n`, bwill be in AP.
Now, `b = (n + 2) th` term `= a+ (n + 2- 1) d`
`therefore d= ((b-a)/n+1)` [Remember] [where, d =common difference] ... (i)

`therefore` `A_1 = a+ d, = a+ 2d, .... , A_n =a+ nd`
`=> A_1 + a+((b-a)/(n+1)) , A_2 = (a+2) ((b-a)/(n+1)) , ........... , A_n = a+n((b-a)/(n+1))`

`text(Corollary I)` The sum of n AM's between two given quantities is equal to
n times the AM between them.
Let two numbers be a and b and `A_1 , A_2 , A_3 , .. , A_n` are n AM's between them.
Then, a, `A_1 , A_2 , A_3 , ... , A_n` , b will be in AP.
`therefore` Sum of n AM's between a and b
`= A_1 + A_2 + A_3 + ........... + A_n`
`= n/2(A_1 + A_2)` [ ` ∵ A_1 + A_2 + A_3 + ........... + A_n` are in AP]
`= n/2(a+d + a+ nd)`
`= n/2[2a+(n+1)d]`
`=n/2(2a + b - a)` [from Eq. (i)]
`= n ((a+b)/2) = n` [AM between a and b] [Remember]

`text(I. Aliter)`
`A_1 + A_2 + A_3 + ... +A_n=(a+ A_1 + A_2 + A_3 + ... +A_n +b)- (a+ b)`
`=(n+2)/2 (a+b) - (a-b)`
`= n ((a+b)/2) = n` [AM of a and b]

`text(II. Aliter)` [This method is applicable only when n is even]
`A_1 + A_2 + A_3 + · ..... + A_(n - 2) + A_(n - 1) + A_n`
`= n (A_1 +A_n)+ (A_2 +A_(n -1)) + (A_3 + A_(n- 2)) + ... upto `n/2` terms
`=(a+ b)+ (a+ b)+ (a+ b)+ ... upto`n/2` times `[ ∵ T_n + T'_n =a+ l]`
`= n/2(a+b)= n((a+b)/2) = n` [AM of a and b]

`text(Corollary II)` The sum of m AM's between any two numbers is to the sum of
n AM's between them as m : n.
Let two numbers be a and b.
`therefore` Sum of m AM's between a and b = m [AM of a and b] ... (i)
Similarly, sum of n AM's between a and b = n [AM of a and b] ... (ii)
`therefore` `text(Sum of m AM's)/text(Sum of n AM's) = (m(AMtext( of a and) b))/(n(AMtext( of a and) b)) = m/n`

Geometric Mean

If three terms are in GP, then the middle term is called the Geometric
Mean (or shortly written as GM) between the other two, so if a, b, care in GP,
then b is the GM of a and c.

`(i)` `text(Single GM of n Positive Numbers)`
Let n positive numbers be `a_1, a_2 , a_3, ... , a_n` and G be the GM of these numbers,
then
`G = (a_1a_2a_3 ............. a_n)^(1/n)`
`text(In particular)` Let a and b be two numbers and G be the GM between them,
then a, G, bare in GP.
Hence , `G = srqt(ab) ; a > 0 , b > 0`

`text(Note)`
`1.` If a < 0, b < 0, then `G =- sqrt(ab)`
`2.` If a < 0, b > 0 or a > 0, b < 0, then GM between a and b does not exist.

`(ii)` `text(Insert n-Geometric Mean Between Two Numbers)`
Let a and b be two given numbers and `G_1 , G_2 , G_3 , ... , G_n` are n GM's between
them.

Then, `a, G_1 , G_2 , G_3, ... , G_n, b` will be in GP.
Now, `b=(n+2)th` term `= ar^(n+2-1)`
`therefore ` ` r = (b/a)^(1/(n+1))` [where r =common ratio] [Remember] ... (i)

`therefore` `G_1 = ar ., G_2 = ar^2 , ........... , G_n = ar^n`
`=> G_1 = a(b/a)^(1/(n+1)) , G_2 = a(b/a)^(2/(n+1)) , ........... , G_n = a(b/a)^(n/(n+1))`

`text(Corollary)` The product of n geometric means between a and b is equal to the
nth power of the geometric mean between a and b.
Let two numbers be a and band `G_1 , G_2 , G_3 , .......... , G_n` are n GM's between them.
Then, `a, G_1 , G_2 , G_3 , ... , G_n , b` will be in GP.
`therefore` Product of n GM's between a and b
`= G_1G_2G_3 ......... G_n = (ar) (ar^2) (ar^3)........... (ar^n)`
`=a^n.r(n.(n+1))/2 = a^n . [(b/a) ^(1/(n+1))]^((n(n+1))/2)` [from Eq. (i)]
`= a^n(b/a)^(n/2) = a^(n//2)b^(n//2) = (sqrt(ab))^n`
`= [GM text(a and b)]^n` [Remember]

`text(Aliter)` [This method is applicable only when n is even]
`G_1 G_2 ....... G_(n-2) G_(n-1) G_n = (G_1 G_n) (G_2G_(n-1))(G_3G_(n-2))......... n/2` factors
`= (ab)(ab)(ab)........n/2` factors ` [∵ T_n X T_n' ==a xx l]`
`=(ab)^(n//2) = (sqrt(ab))^n = [ GM text(of a and b)]^n`

Harmonic Mean

If three terms are in HP, then the middle term is called the Harmonic
Mean (or shortly written as HM) between the other two, so if a, b, care in HP,
then b is the HM of a and c.

`(i)` `text(Single HM of n Positive Numbers)`

Let n positive numbers be `a_1, a_2 , a_3, ... , a_n` and H be the HM of these
numbers, then

`H = n/(1/a_1 + 1/a_2 + 1/a_3 + ........ + 1/a_n)`

In particular Let a and b be two given numbers and H be the HM between
them a, H, bare in HP.

Hence , `H = 2/(1/a + 1/b)`

i.e., `H = (2ab)/((a+b))`

`text(Note)`
HM of a,b,c is `3/(1/a+1/b+1/c)` or `(3abc)/(ab + bc+ ca)`

`text(Caution)` The AM between two numbers a and b is `(a+b)/2.` It does not follow that HM
between the same numbers is `2/(a+b).` The HM is the reciprocal of `(1/a + 1/b)/2`
i.e., `(2ab)/(a+b)`

`(ii)` `text(Insert a-Harmonic Mean Between Two Numbers)`

Let a and b be two given numbers and `H_1, H_2 , H_3 , ... , H_n` are n HM's between
them.

Then, `a, H_1 , H_2, H_3 , ... , H_n, b` will be in HP, if D be the common difference of
the corresponding AP.

`therefore b = (n+2)th` term of HP
`= 1/((n+2)th text(term of corresponding AP)) = 1/ (1/a + (n+2-1) D)`
`=> D = (1/b - 1/a)/(n+1)` [Remember]
`therefore 1/H_1 = 1/a + D , 1/H_2 = 1/a + 2D , ......... , 1/H_n = 1/a + nd`
`=> 1/H_1 = 1/a + (a-b)/(ab(n+1)) , 1/H_2 = 1/a + (2(a-b))/(ab(n+1)) , ........ , 1/H_n = 1/a + (n(a-b))/(ab(n+1))`

`text(Corollary)` The sum of reciprocals of n harmonic means between two given
numbers is n times the reciprocal of single HM between them.
Let two numbers be a and band `H_1,H_2,H_3 , ... ,H_n` are n HM's between
them. Then, `a, H_1 , H_2 , H_3 , ... , H_n , b` will be in HP.
`therefore 1/h_1 + 1/H_2 + 1/H_3 + ............ + 1/H_n = n/2(1/H_1 + 1/H_n)` `[∵ S_n = n/2(a+l)]`
`= n/2(1/a + D + 1/b -D) = n/2(1/a + 1/b)`
`=n/(2/(1/a+1/b)) = n/(HM text( of a and b))`

`text(Aliter)` [This method is applicable only when n is even]
`1/H_1 + 1/H_2 + 1/H_3 + ........ + 1/H_(n-2) + 1/H_(n-1) + 1/H_n`
`(1/H_1 + 1/H_n ) + (1/H_2 + 1/H_(n-1)) + ................. text(upto) n/2 ` terms
`= (1/a + D + 1/b - D) + (1/a + 2D + 1/b - 2D) + (1/a + 3D + 1/b - 3D) + ............ + text(upto) n/2 text(terms)`
`(1/a + 1/b) + (1/a + 1/b) + (1/a + 1/b) + ............ + text(upto) n/2 text(terms)`
`= n/2(1/a+1/b)=n/(2/(1/a+1/b)) = n/(HM text(os a and b))`

Theorems

`text(Theorem 1)`

`text(Relation between `A.M, G.M.` and `H.M` :)`

If `a` and `b` are two positive numbers then `A ge G ge H` and `A, G, H` are in `GP. ` i.e. `G^2=AH`

`text(proof :)` We have `A=(a+b)/2,G=sqrt(ab)` and `H=(2ab)/(a+b)`

now `AH = ab = G^2 => AGH` are in `GP.`

also `A/G=G/H; :. A ge G => G ge H`

Hence `A ge G ge H` Infact `AM ge GM ge HM`

`text( Theorem 2 :)`

If A, a, H are arithmetic, geometric and harmonic means of th:ree given
numbers a, band c, then the equation having a, b, cas its roots is

`x^2 - 3Ax^2 + (3G^2)/Hx - G^3 =0 ` [Remember]

`text(Proof)` `∵ A = AM of a.b,c = (a+b+c)/3`
i.e., `a+b+c = 3A..................(i)`
`G = GM text(of a,b,c) = (abc)^(1//3)`
i.e., `abc = G^3...................(ii)`
and `H = HM` of a ,b ,c
`= 3/(1/a+1/b+1/c) = (3abc)/(ab+bc+ca)`
`(3G^3)/(ab+bc+ca)` [from Eq. (ii)]
i.e., `ab+bc+ca = (3G^3)/H`
`therefore a,b,c` are the roots of equation
`x^3 - (a+b+c)x^2 + (ab+bc+ ca)x - abc =0`
i.e., `x^3 - 3Ax^2 + (3G^3)/H - G^3 =0` [from Eqs. (i), (ii) and (iii)]