Arithmetic Mean

• The A.M. between two numbers a and b is
          A =(a +b)/2
• Similarly, if a, A₁, A₂, ..., A_n, b are in A.P., then A₁, A₂, ..., A_n are called n arithmetic means between a and b.

Illustrative Examples

Example

Insert six arithmetic means between 2 and 16. Also prove that their sum is 6 times the A.M. between 2 and 16.

Solution

Let A₁, A₂, ..., A₆ be six A.M.s between 2 and 16. Then, by def., 2, A₁, A₂, ..., A₆, 16 are in A.P.
Let d be the common difference. Here 16 is the 8th term.
16 = 2 +7d => d = 2.
Hence six A.M.s are a +d, a +2d, ..., a +6d
      i.e. 4, 6, 8, 10, 12, 14.
Now sum of these means
      = 6(4 +14)/2 = 54 = 6.9 = 6(2 +16)/2
     = 6 times the A.M. between 2 and 16.

Example

If A is the A.M. between a and b, show that
            (A +2 a)/(A -b) + (A +2 b)/(A -a) = 4           

Solution

Since A is the A.M. between a and b, A = (a +b)/2
Hence (A +2 a)/(A -b) + (A +2 b)/(A -a)
        =
= (5a + b)/(a - b) + (5b + a)/(b - a) = (4a - 4b)/(a-b)
= 4

Exercise

1. (i) Insert 5 arithmetic means between -2 and 10. Show that their sum is 5 times the arithmetic mean between -2 and 10.
   (ii) Insert 4 arithmetic means between 12 and -3.
   (iii) Insert 10 A.M.s between -5 and 17 and prove that their sum is 10 times the A.M. between -5 and 17.
2. If A is the A.M. between a and b, prove that
   (i) (A -a)² +(A -b)² = (a -b)²/2
   (ii) 4(a -A)(A -b) = (a -b)².
3. The ratio of the first to the last of n A.M.s between 5 and 35 is 1 : 4. Find n.
4. There are n arithmetic means between 3 and 17. The ratio of the first mean to the last mean is 1 : 3. Find n.
5. If the A.M. between p th and q th items of an A.P. is equal to the A.M. between rth and sth items, show that p +q = r +s.

Answers