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Question
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
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Solution
Let the terms of the given G.P. be \[\frac{a}{r}, \text { a and ar }\]
∴ Product = 216
\[\Rightarrow a^3 = 216\]
\[ \Rightarrow a = 6\]
It is given that \[\frac{a}{r} + 2, a + 8 \text { and ar } + 6\] are in A.P.
\[\therefore 2\left( a + 8 \right) = \frac{a}{r} + 2 + ar + 6\]
\[\text { Putting a = 6, we get }\]
\[ \Rightarrow 28 = \frac{6}{r} + 2 + 6r + 6\]
\[ \Rightarrow 28r = 6 r^2 + 8r + 6\]
\[ \Rightarrow 6 r^2 - 20r + 6 = 0\]
\[ \Rightarrow \left( 6r - 2 \right)\left( r - 3 \right) = 0\]
\[ \Rightarrow r = \frac{1}{3}, 3\]
\[\text { Hence, putting the values of a and r, the required numbers are 18, 6, 2 or 2, 6 and 18 }.\]
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