Permutations and Combinations

Permutations and Combinations

Question

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


Answer

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
                                    
                    

More Chapters from Permutations and Combinations