If a, b, c, d are in G.P., then prove that  are in G.P.

Asked by Topperlearning User | 25th Sep, 2017, 12:20: PM

Expert Answer:

Here, a, b, c, d are in G.P. Let r be the common ratio.

  b = ar, c =, d =
Also,
     

begin mathsize 12px style LHS space equals space fraction numerator straight b squared space minus space straight c squared over denominator straight a squared space minus space straight b squared end fraction
space space space space space space space space space equals space fraction numerator open parentheses ar close parentheses squared space minus space open parentheses ar squared close parentheses squared over denominator straight a squared space minus space open parentheses ar close parentheses squared end fraction
space space space space space space space space space equals fraction numerator straight a squared straight r squared space minus space straight a squared straight r to the power of 4 over denominator straight a squared space minus space straight a squared straight r squared end fraction
space space space space space space space space space equals fraction numerator space straight a squared straight r squared space open parentheses 1 minus straight r squared close parentheses over denominator straight a squared space open parentheses open parentheses 1 minus straight r squared close parentheses close parentheses end fraction
space space space space space space space space equals space straight r squared
Similarly comma
RHS space equals space fraction numerator straight c squared space minus space straight a squared over denominator straight b squared space minus space straight c squared end fraction
space space space space space space space space space equals fraction numerator open parentheses ar squared close parentheses squared space minus space open parentheses ar cubed close parentheses squared over denominator open parentheses ar close parentheses squared space minus space open parentheses ar squared close parentheses squared end fraction
space space space space space space space space space equals space straight r squared end style

     
Hence, it is a G.P.

Answered by  | 25th Sep, 2017, 02:20: PM