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Question
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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Solution
Let the terms of the G.P be \[\frac{a}{r},\text { a and ar .}\]
∴ Product of the G.P. = 1
\[\Rightarrow a^3 = 1\]
\[ \Rightarrow a = 1\]
Now, sum of the G.P. = \[\frac{39}{10}\]
\[\Rightarrow \frac{a}{r} + a + ar = \frac{39}{10}\]
\[ \Rightarrow a\left( \frac{1}{r} + 1 + r \right) = \frac{39}{10}\]
\[ \Rightarrow 1\left( \frac{1}{r} + 1 + r \right) = \frac{39}{10}\]
\[ \Rightarrow 10 r^2 + 10r + 10 = 39r\]
\[ \Rightarrow 10 r^2 - 29r + 10 = 0\]
\[ \Rightarrow 10 r^2 - 25r - 4r + 10 = 0\]
\[ \Rightarrow 5r(2r - 5) - 2(2r - 5) = 0\]
\[ \Rightarrow \left( 5r - 2 \right)\left( 2r - 5 \right) = 0\]
\[ \Rightarrow r = \frac{2}{5}, \frac{5}{2}\]
Hence, putting the values of a and r , the required numbers are \[\frac{5}{2}, 1, \frac{2}{5} \text { or } \frac{2}{5}, 1 \text { and }\frac{5}{2}\].
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