Examples

Geometric sequence: revealing the format

Find $a_0$ and $r$ and $a_n$ (written in the geometric sequence format) for the following geometric sequences:

(a) $a_n={(-{\displaystyle \frac{1}{2}})}^n$ $$ (b) $b_n=-3\left({\displaystyle \frac{2^{n+1}}{5^n}}\right)$ $$
(c) $c_n=e^{5-7n}$

Solution

(a)

Plug in $n=0$ to obtain $a_0=1$. Notice that $a_{n+1}/a_n=-1/2$ and so therefore $r=-1/2$. Then the ‘general term’ is $a_n=a_0\cdot r^n=1\cdot {(-1/2)}^n$.

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(b)

Rewrite the fraction:

$$\frac{2^{n+1}}{5^n}\gg \gg 2\cdot {\left(\frac{2}{5}\right)}^n$$

Plug that in and observe $b_n=-6\cdot {(2/5)}^n$. From this format we can read off $b_0=-6$ and $r=2/5$.

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(c)

Rewrite:

$$c_n\gg \gg e^5\cdot e^{-7n}\gg \gg e^5\cdot {\left(e^{-7}\right)}^n$$

From this format we can read off $c_0=e^5$ and $r=e^{-7}$.