Show that the sequence defined by an = \(\frac{2}{3^n}\), n∈N is a

1.	Show that the sequence defined by an = \(\frac{2}{3^n}\), n∈N is a G.P.
Answer» Put n = 1,2,3,4… a₁, a₂, a₃, a₄ _{\(\Rightarrow\)}a₁=\(\frac{2}{3}\), a₂ = \(\frac{2}{9}\), a₃ = \(\frac{2}{27}\), a₄ = \(\frac{2}{81}\), ..... If a_1, a₂, a₃...... \(\Rightarrow\) (a₂)² = a₁,a₃ \(\Rightarrow\) \((\frac{2}{9})^2\)=\(\frac{2}{3}\)=\(\frac{2}{27}\) \(\Rightarrow\) \(\frac{4}{81}\) So, It is GP. Common ratio = r= \(\cfrac{\frac{1}{3}}{\frac{1}{2}}\) = \(\frac{2}{3}\)

Show that the sequence defined by an = \(\frac{2}{3^n}\), n∈N is a G.P.

Answer»

Put n = 1,2,3,4…

a₁, a₂, a₃, a₄

_{\(\Rightarrow\)}a₁=\(\frac{2}{3}\), a₂ = \(\frac{2}{9}\), a₃ = \(\frac{2}{27}\), a₄ = \(\frac{2}{81}\), .....

If a_1, a₂, a₃......

\(\Rightarrow\) (a₂)² = a₁,a₃

\(\Rightarrow\) \((\frac{2}{9})^2\)=\(\frac{2}{3}\)=\(\frac{2}{27}\)

\(\Rightarrow\) \(\frac{4}{81}\)

So, It is GP.

Common ratio = r= \(\cfrac{\frac{1}{3}}{\frac{1}{2}}\) = \(\frac{2}{3}\)