Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Solution
Let the required numbers be \[\frac{a}{r}, \text { a and ar } .\]
Product of the G.P. = 729
\[\Rightarrow a^3 = 729\]
\[ \Rightarrow a = 9\]
Sum of the products in pairs = 819
\[\Rightarrow \frac{a}{r} \times a + a \times ar + ar \times \frac{a}{r} = 819\]
\[ \Rightarrow a^2 \left( \frac{1}{r} + r + 1 \right) = 819\]
\[ \Rightarrow 81\left( \frac{1 + r^2 + r}{r} \right) = 819\]
\[ \Rightarrow 9\left( r^2 + r + 1 \right) = 91r\]
\[ \Rightarrow 9 r^2 - 82r + 9 = 0\]
\[ \Rightarrow 9 r^2 - 81r - r + 9 = 0\]
\[ \Rightarrow \left( 9r - 1 \right)\left( r - 9 \right) = 0\]
\[ \Rightarrow r = \frac{1}{9}, 9\]
\[\text { Hence, putting the values of a and r, we get the numbers to be 81, 9 and 1 or 1, 9 and 81 } .\]
