The sum of the first three terms of a G.P.

Question

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

Accepted Answer

Let three terms of G.P. are $${a \over r}$$, a, ar

$$ \therefore $$ $$a\left( {{1 \over r} + 1 + r} \right)$$ = S ...(1)

and a3
 = 27

$$ \Rightarrow $$ a = 3

$$ \therefore $$ $$3\left( {{1 \over r} + 1 + r} \right)$$ = S

$$ \Rightarrow $$ $${{1 \over r} + r = {S \over 3} - 1}$$

$$ \Rightarrow $$ As $${{1 \over r} + r \ge 2}$$ or $${{1 \over r} + r \le  - 2}$$

$$ \therefore $$ $${{S \over 3} - 1 \ge 2}$$ or $${{S \over 3} - 1 \le  - 2}$$

$$ \Rightarrow $$ $${{S \over 3} \ge 3}$$ or $${{S \over 3} \le  - 1}$$

$$ \Rightarrow $$ S $$ \ge $$ 9 or S$$ \le $$ -3

$$ \therefore $$ S $$ \in $$ (-$$ \propto $$, -3] $$ \cup $$ [9, $$\infty $$)

Solutions · 1