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  4. If x, 2x+2, 3x+3,..... are in GP, then the fourth term is

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Q. If $ x,\,2x+2,\,\,3x+3,..... $ are in GP, then the fourth term is

J & K CETJ & K CET 2003

Solution:

Since, given series is in GP.
$ \therefore $ $ {{(2x+2)}^{2}}=x\times (3x+3) $
$ \Rightarrow $ $ 4{{x}^{2}}+4+8x=3{{x}^{2}}+3x $
$ \Rightarrow $ $ {{x}^{2}}+5x+4=0 $
$ \Rightarrow $ $ x=\frac{-5\pm \sqrt{25-16}}{2} $
$ \Rightarrow $ $ x=\frac{-5\pm 3}{2} $
$ \Rightarrow $ $ x=-\frac{8}{2}=-4 $ and $ x=-\frac{2}{2}=-1 $ At $ x=-1, $
second terms become zero so we neglect that.
At $ x=-4,\,\,a=-4,\,\,\,r=3/2 $
$ \therefore $ $ {{T}_{4}}=-4\times {{(3/2)}^{3}}=-\frac{27}{2}=-13.5 $