Question

If a,b,c are in AP, show that
`1/((sqrtb+sqrtc)),1/((sqrtc+sqrta)),1/((sqrta+sqrtb))` are in AP.

Answer 1

Since, a,b,c are in AP, we have.
Now, `1/((sqrtb+sqrtc)),1/((sqrtc+sqrta)),1/((sqrta+sqrtb))` will be in AP.
if `1/((sqrtc+sqrta))-1/((sqrtb+sqrtc))=1/((sqrta+sqrtb))-1/((sqrtc+sqrta))`
i.e, if `((sqrtb+sqrtc)-(sqrtc+sqrta))/((sqrtc+sqrta) (sqrtb+sqrtc))=((sqrtc+sqrta)-(sqrta+sqrtb))/((sqrta+sqrta+sqrtb) (sqrtc+sqrta))`
i.e, if`((sqrtb-sqrta))/((sqrtc+sqrta)(sqrtb+sqrtc))=((sqrtc-sqrtb))/((sqrta+sqrtb)(sqrtc+sqrta))`
i,e., if ` ((sqrtb-sqrta))/((sqrtb+sqrtc))=((sqrtc-sqrtb))/((sqrta+sqrtb))`
i.e, if b-a =c-b
i.,e if 2b = a+c which is true by(i)
a,b,c, are in AP, ` Rightarrow 1/((sqrtb +sqrtc)) ,1/((sqrtc+sqrta)) , 1/((sqrta + sqrtb))` are in AP.