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Question
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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Solution
\[\text { Given }: \]
\[\text { First term, } a = 5 \]
\[\text { Common ratio }, r = 2\]
\[ a_n = \left( 5 \right) \left( 2 \right)^{n - 1} . . . \left( 1 \right)\]
\[\text { Similarly, } a_n = \left( 1280 \right) \left( \frac{1}{2} \right)^{n - 1} . . . \left( 2 \right)\]
\[\text { From }\left( 1 \right) \text { and } \left( 2 \right)\]
\[\left( 5 \right) \left( 2 \right)^{n - 1} = \left( 1280 \right) \left( \frac{1}{2} \right)^{n - 1} \]
\[ \Rightarrow \frac{1}{256} = \left( \frac{1}{4} \right)^{n - 1} \]
\[ \Rightarrow \left( \frac{1}{4} \right)^4 = \left( \frac{1}{4} \right)^{n - 1} \]
\[ \Rightarrow n - 1 = 4 \]
\[ \Rightarrow n = 5\]
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