Question
Let $$S$$ be the sum, $$P$$ be the product and $$R$$ be the sum of the reciprocals of 3 terms of a G.P. Then $${P^2}{R^3}:{S^3}$$ is equal to
A.
$$1 : 1$$
B.
$${ {{\text{first term}}}} : 1$$
C.
$${\left( {{\text{first term}}} \right)^2}:{\left( {{\text{common ratio}}} \right)^2}$$
D.
$${\left( {{\text{common ratio}}} \right)^n}:1$$
Answer :
$$1 : 1$$
Solution :
If the three terms of the G.P. be $$\frac{a}{r},a\,{\text{and }}ar$$ then
$$\eqalign{
& S = \frac{a}{r} + a + ar = \frac{a}{r}\left( {1 + r + {r^2}} \right) \cr
& P = {a^3}\,{\text{and }}R = \frac{r}{a} + \frac{1}{a} + \frac{1}{{ar}} = \frac{1}{{ar}}\left( {{r^2} + r + 1} \right) \cr
& {\text{Now,}}\frac{{{P^2}{R^3}}}{{{S^3}}} = \frac{{{a^6}\frac{1}{{{a^3}{r^3}}}{{\left( {{r^2} + 2 + 1} \right)}^3}}}{{\frac{{{a^3}}}{{{r^3}}}{{\left( {{r^2} + r + 1} \right)}^3}}} = 1 \cr} $$
So, the required ratio is $$1 : 1 .$$