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+1 vote
73.2k views
in Mathematics by (43.7k points)

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

(1) [–3, ∞) 

(2) (– ∞, 9] 

(3) (–∞, –9] ∪ [3, ∞)

(4) (-∞, –3] ∪[9, ∞)

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3 Answers

+2 votes
by (48.5k points)

Answer is (4) (-∞, –3] ∪[9, ∞)

Let three terms of G.P. are \(\frac{a}{r}\),a,ar

product = 27

⇒ a3 = 27 ⇒ a = 3

+1 vote
by (80 points)

\(\mathsf{the\:three\:terms\:GP\:are\:\dfrac{a}{R},a,ar}\)

\(\mathsf{product\:of\:three\:numbers\:are\:a^3=27}\)

                                →   \(\mathsf{a=3}\)............................(1)

\(\mathsf{substitute\: in\:1} \)

          →\(\mathsf{\dfrac{3}{r}+3r+3=S}\)

\(\mathsf{here\:r>0}\)\

         →\(\mathsf{\dfrac{\dfrac{3}{r}+3r}{2}}>\sqrt{3^2}\)

\(\mathsf{\dfrac{3}{r}+3r \geq 6}\)............................(2)

\(\mathsf{for\: r<0}\)

     \(\mathsf{\dfrac{3}{r}+3r \leq -6}\).........................(3)

\(\mathsf{from\:2\:and\:3\:we\:get}\)

so option is 4

0 votes
by (20 points)

Hence the correct answer is option (4)

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