Question

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

Answer 1

Let the terms of the the given G.P. be

\[\frac{a}{r}, \text { a and ar } .\]

Then, product of the G.P. = 3375

\[\Rightarrow\] a³ = 3375

\[\Rightarrow\] a = 15
Similarly, sum of the G.P. = 65

\[\Rightarrow \frac{a}{r} + a + ar = 65\]

Substituting the value of a

\[\frac{15}{r} + 15 + 15r = 65\]

\[ \Rightarrow 15 r^2 + 15r + 15 = 65r\]

\[ \Rightarrow 15 r^2 - 50r + 15 = 0\]

\[ \Rightarrow 5\left( 3 r^2 - 10r + 3 \right) = 0\]

\[ \Rightarrow 3 r^2 - 10r + 3 = 0\]

\[ \Rightarrow \left( 3r - 1 \right)\left( r - 3 \right) = 0\]

\[ \Rightarrow r = \frac{1}{3}, 3\]

Hence, the G.P. for a = 15 and r = \[\frac{1}{3}\] is 45, 15, 5.

And, the G.P. for a = 15 and r = 3 is 5, 15, 45.