The mth term of a H.P is `n` and the nth term is `m` . Proves that its rth term is `m n//rdot`

Explanation

Let the H.P. be `1/a,1/(a+d),1/(a+2d),…`
Then, `a_(m)=nanda_(n)=m`
`rArr1/(a+(m-1)d)=nand1/(a+(n-1)d)=m`
`a+(m-1)d=1/n and a+(n-1)d=1/m`
`rArr{a+(m-1)d}-{a+(n-1)d}=1/n-1/m`
[On subtracting]
`rArr(m-n)d=(m-n)/(mn)`
`rArrd=1/(mn)`
Putting d=`1/(mn)` in a+(m-1)d=`1/n`, we get
`a+((m-1))/(mn)=1/n`
or `a+1/n-1/(mn)=1/n`
or `a=1/(mn)`
Now, `a_(r)=1/(a+(r-1)d)`
`=1/(1/(mn)+((r-1))/(mn))`
`=(mn)/(1+r-1)`
`=(mn)/r`