Permutations and Combinations

Permutations and Combinations

Question

If a, b, c, d are in GP., show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2

Answer

Here, a, b, c, d are in G.P. Let r be the common ratio of the G.P.
∴       b = ar, c = ar2, d = ar3
Now,       left parenthesis straight a squared plus straight b squared plus straight c squared right parenthesis space left parenthesis straight b squared plus straight c squared plus straight d squared right parenthesis space equals space left parenthesis ab plus bc plus cd right parenthesis squared
if           left parenthesis straight a squared plus straight a squared straight r squared plus straight a squared straight r to the power of 4 right parenthesis space left parenthesis straight a squared straight r squared plus straight a squared straight r to the power of 4 plus straight a squared straight r to the power of 6 right parenthesis space equals space left parenthesis straight a. space ar space plus space ar. space ar squared space plus space ar squared. space ar cubed right parenthesis squared
or  if        straight a squared left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis straight a squared straight r squared space left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis space equals space left parenthesis straight a squared straight r plus straight a squared straight r cubed plus straight a squared straight r to the power of 5 right parenthesis squared
or if      space space space space space straight a to the power of 4 straight r squared left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis squared space equals space left square bracket straight a squared straight r left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis right square bracket squared
or if          straight a to the power of 4 straight r squared left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis squared space equals space straight a to the power of 4 straight r squared left parenthesis 1 plus straight r squared plus straight r to the power of 4 right parenthesis squared
which is true.
Hence, if a, b, c d are in G.P. then left parenthesis straight a squared plus straight b squared plus straight c squared right parenthesis space left parenthesis straight b squared plus straight c squared plus straight d squared right parenthesis space equals space left parenthesis ab plus bc plus cd right parenthesis squared

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