# The sum of the first three term of GP .is S and their product is 27 . then all such S lie in :

Solution: Let a is first term and r is the common ratio

Now,

$\\ a/r\times a\times ar=27\\ \\\Rightarrow a=3\\\\\Rightarrow a/r+a+ar=S\\ \\\Rightarrow 1/r+1+r=S/3\\ \\\Rightarrow r+1/r=S/3-1$

$\\(r+1/r)\geq 2;or;(r+1/r)\leq -2\\ \\\Rightarrow S/3\geqslant 3;or S/3\leq -1\\ \\\Rightarrow S\geq 9;or;S\leq -1$

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