If `a,b,c` are in AP and p is the AM between a and b and q is the AM between b and c, then show that b is the AM between p and q.

Explanation

`therefore a,b,c` are inAP.
`therefore " " 2b=a+c " " "…….(i)"`
`therefore` p is the AM between a and b.
`therefore " " p=(a+b)/(2) " " "……..(ii)"`
`therefore` q is the AM between b and c.
`therefore " " q=(b+c)/(2) " " "......(iii)"`
On adding Eqs. (ii) and (iii), than
`p+q=(a+b)/(2)+(b+c)/(2)=(a+c+2b)/(2)=(2b+2b)/(2) [" using Eq. (i)"]`
`therefore p+q=2b " or " b=(p+q)/(2)`
Hence, b is the AM between p and q.