Permutations and Combinations

Permutations and Combinations

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.

Answer
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
if      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
if        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.

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