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`
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`