Permutations and Combinations

If a, b, c are in A.P., prove that are also in A.P. - Wired Faculty

Question

If a, b, c are in A.P., prove that $fraction numerator 1 over denominator square root of straight b plus square root of straight c end fraction comma space fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction comma space fraction numerator 1 over denominator square root of straight a plus square root of straight b space end root end fraction$ are also in A.P.

Solution

fraction numerator 1 over denominator square root of straight b plus square root of straight c end fraction comma space fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction comma space fraction numerator 1 over denominator square root of straight a plus square root of straight b end fraction

will be in A.P.
if

fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction minus fraction numerator 1 over denominator square root of straight b plus square root of straight c end fraction space equals space fraction numerator 1 over denominator square root of straight a plus square root of straight b end fraction minus fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction

fraction numerator square root of straight b plus square root of straight c minus square root of straight c minus square root of straight a over denominator left parenthesis square root of straight c plus square root of straight a right parenthesis space left parenthesis square root of straight b plus square root of straight c right parenthesis end fraction space equals space fraction numerator square root of straight c plus square root of straight a minus square root of straight a minus square root of straight b over denominator left parenthesis square root of straight a plus square root of straight b right parenthesis space left parenthesis square root of straight c plus square root of straight a right parenthesis end fraction

or if

fraction numerator square root of straight b minus space square root of straight a over denominator square root of straight c plus square root of straight b end fraction space equals space fraction numerator square root of straight c minus square root of straight b over denominator square root of straight b plus square root of straight a end fraction

left parenthesis square root of straight b minus square root of straight a right parenthesis space left parenthesis square root of straight b plus square root of straight a right parenthesis space equals space left parenthesis square root of straight c minus space square root of straight b right parenthesis space left parenthesis square root of straight c plus square root of straight b right parenthesis

if b - a = c - b
if a, b, c are in A.P. which is true.
Hence,

fraction numerator 1 over denominator square root of straight b plus square root of straight c end fraction comma space fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction comma space fraction numerator 1 over denominator square root of straight a plus square root of straight b end fraction

are in A.P.