Question

If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{First}\mathrm{term},a=5 \\ \mathrm{Common}\mathrm{ratio},r=2 \\ {a}_n=\left(5\right){\left(2\right)}^{n-1}...\left(1\right) \\ \mathrm{Similarly},a_n=\left(1280\right){\left(\frac{1}{2}\right)}^{n-1}...\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right) \\ \left(5\right){\left(2\right)}^{n-1}=\left(1280\right){\left(\frac{1}{2}\right)}^{n-1} \\ \Rightarrow \frac{1}{256}={\left(\frac{1}{4}\right)}^{n-1} \\ \Rightarrow {\left(\frac{1}{4}\right)}^4={\left(\frac{1}{4}\right)}^{n-1} \\ \Rightarrow n-1=4 \\ \Rightarrow n=5 \end{gathered}$$