Write an expression for the nth term of the sequence. \frac{5}{2}, \frac{5}{4}, \frac{5}{8},...

Given:

The geometric sequence:

$$\displaystyle \frac{5}{2},\frac{5}{4},\frac{5}{8},.............. $$

Here

{eq}\text{First term} (a) = \displaystyle \frac{5}{2} {/eq}

and

{eq}\displaystyle \text{Common ratio} (r) = \frac{\displaystyle \frac{5}{4}}{\displaystyle \frac{5}{2}} = \frac{\displaystyle \frac{5}{8}}{\displaystyle \frac{5}{4}} = \frac{1}{2} {/eq}

Now the {eq}n {/eq}th term of the given geometric sequence:

$$\begin{align} a_{n} & = a.r^{n-1} \\[0.2 cm] & = \frac{5}{2}.\left(\frac{1}{2}\right)^{n-1} \\[0.2 cm] & = \frac{5}{2} \times \frac{1}{2^{n-1} } \\[0.2 cm] a_{n} & = \frac{5}{2^{n} } \\[0.2 cm] \end{align} $$