Permutations And Combinations

  • Question
    CBSEENMA11013144

    If a, b, c are in A.P. and x,y,z are in G.P., show that 

    straight x to the power of straight b minus straight c end exponent. space straight y to the power of straight c minus straight a end exponent. space straight z to the power of straight a minus straight b end exponent space equals space 1


    Solution

    Since a, b, c are in A.P.
    ∴              b - a = c - b rightwards double arrow2b = c + a
    Here, x, y, z are in G.P.
    Let r be the common ratio
    ∴               y = xr, z = xr2
    Now, space space space space space space straight x to the power of straight b minus straight c end exponent. space straight y to the power of straight c minus straight a end exponent. space straight z to the power of straight a minus straight b end exponent space equals space straight x to the power of straight b minus straight c end exponent left parenthesis xr right parenthesis to the power of straight c minus straight a end exponent. space left parenthesis xr squared right parenthesis to the power of straight a minus straight b end exponent space equals space straight x to the power of straight b minus straight c end exponent. space straight x to the power of straight c minus straight a end exponent. space straight r to the power of straight c minus straight a end exponent. space straight x to the power of straight a minus straight b end exponent. space straight r to the power of 2 straight a minus 2 straight b end exponent.
                                        equals space straight x to the power of straight b minus straight c plus straight c minus straight a plus straight a minus straight b end exponent. straight r to the power of straight c minus straight a plus 2 straight a minus 2 straight b end exponent space equals space straight x to the power of 0. straight r to the power of straight c plus straight a minus 2 straight b end exponent
                                         equals straight x to the power of 0. space straight r to the power of 2 straight b minus 2 straight b end exponent space equals space 1. space straight r to the power of 0                                       [By using (i)]
                                          = 1 . 1 = 1
    ∴    straight x to the power of straight b minus straight c end exponent. straight y to the power of straight c minus straight a end exponent. straight z to the power of straight a minus straight b end exponent equals space 1
                                        
                        

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