Question
Wed March 14, 2012 By:

Q;if a1,a2,a3....an are positive numbers in a.p. then show that 1/root(a1)+root(a2) +1/root(a2)+root(a3) + 1/root(a3)+root(a4).....+ 1/root(an)+root(an+1)= n/root(a1)+root(an+1)

Expert Reply
Thu March 15, 2012
let d be the common difference in the given AP
So if we take the first term
1/{root(a1) + root(a2)} and rationalize the denominator taht is multiply the numerator and the denominator by root(a2) - root(a1)
we get ,
{root(a2) - root(a1)}/(a2-a1) = {root(a2) - root(a1)}/d
 
Similarly 1/{root(a2) + root(a3)} = {root(a3) - root(a2)}/d
 
On adding all the terms we get
 
{root(a2) - root(a1) + root(a3) - root(a2) + ......... + root(an+1) - root(an)}/d
 
= {root(an+1) - root(a1)}/d        .........1
 
Also d = {an+1 - a1}/n = {(root(an+1) - root(a1))(root(an+1) + root(a1))}/n
 
On substituting the value of d in eq1 we get
R.H.S.
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