Question

If a1,a2,a3,....,an are in ap,ai>0 for all i,show that 1/(√a1+√a2)+1/(√a2+√a3)+.....+1/(√an-1 + √an)=(n-1)/(√a1+√an)

Answer 1

1/(√a1+√a2)+1/(√a2+√a3)+.....+1/(√an-1 +√an)(√a2-√a3)/(a2-a3)+...(√an-1-√an)/({an-1}-an)...[rationalising]=(√a1-√a2)/(a1-a2)+(√a2-√a3)/d...-(√an-1-√an)/d...[(tn)-(tn-1)=d=comn diff]=-(√a1-√a2)/d-all the conecutive term willcancel excp√an)/d=-(a1-an)/(√a1+√an)d=-(√a1-=-(a1-a1-(n-1)d)/(√a1+√an)d=(n-1)/(√a1+√an)on