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Question
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
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Solution
\[\text { Let the 5 G . M . s betweem 16 and } \frac{1}{4} \text { be } G_1 , G_2 , G_3 , G_4 \text { and } G_5 . \]
\[16, G_1 , G_2 , G_3 , G_4 , G_5 , \frac{1}{4}\]
\[ \Rightarrow a = 16, n = 7 \text { and } a_7 = \frac{1}{4}\]
\[ \because a_7 = \frac{1}{4}\]
\[ \Rightarrow a r^6 = \frac{1}{4}\]
\[ \Rightarrow r^6 = \frac{1}{4 \times 16}\]
\[ \Rightarrow r^6 = \left( \frac{1}{2} \right)^6 \]
\[ \Rightarrow r = \frac{1}{2}\]
\[ \therefore G_1 = a_2 = ar = 16\left( \frac{1}{2} \right) = 8\]
\[ G_2 = a_3 = a r^2 = 16 \left( \frac{1}{2} \right)^2 = 4\]
\[ G_3 = a_4 = a r^3 = 16 \left( \frac{1}{2} \right)^3 = 2\]
\[ G_4 = a_5 = a r^4 = 16 \left( \frac{1}{2} \right)^4 = 1\]
\[ G_5 = a_6 = a r^5 = 16 \left( \frac{1}{2} \right)^5 = \frac{1}{2}\]
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