Geometric Sequences

What is a geometric sequence?

How is the common ratio of a geometric sequence found?

What is the procedure for determining whether a sequence is geometric?

What is the difference between an arithmetic sequence and a geometric sequence?

Describe how exponential functions and geometric sequences are similar. How are they different?

$1,3,9,27,81,\mathrm{...}$

$-0.125,0.25,-0.5,1,-2,\mathrm{...}$

$-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128},\mathrm{...}$

$-6,-12,-24,-48,-96,\mathrm{...}$

$5,5.2,5.4,5.6,5.8,\mathrm{...}$

$-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},\mathrm{...}$

$6,8,11,15,20,\mathrm{...}$

$0.8,4,20,100,500,\mathrm{...}$

$\begin{array}{cc}{a}_1=8,& r=0.3\end{array}$

$\begin{array}{cc}{a}_1=5,& r=\frac{1}{5}\end{array}$

$\begin{array}{cc}{a}_7=64,& a_{10}\end{array}=512$

$\begin{array}{cc}{a}_6=25,& a_8\end{array}=6.25$

The first term is $\mathrm{2,}$ and the common ratio is $3.$ Find the 5^th term.

The first term is 16 and the common ratio is $-\frac{1}{3}.$ Find the 4^th term.

$a_n=\left\{-1,2,-4,8,\mathrm{...}\right\}.$ Find $a_{12}.$

$a_n=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},\mathrm{...}\right\}.$ Find $a_7.$

$\begin{array}{cc}{a}_1=-486,& a_n=-\frac{1}{3}\end{array}{a}_{n-1}$

$\begin{array}{cc}{a}_1=7,& a_n=0.2a_{n-1}\end{array}$

$a_n=\left\{-1,5,-25,125,\mathrm{...}\right\}$

$a_n=\left\{-32,-16,-8,-4,\mathrm{...}\right\}$

$a_n=\left\{14,56,224,896,\mathrm{...}\right\}$

$a_n=\left\{10,-3,0.9,-0.27,\mathrm{...}\right\}$

$a_n=\left\{0.61,1.83,5.49,16.47,\mathrm{...}\right\}$

$a_n=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},\mathrm{...}\right\}$

$a_n=\left\{-2,\frac{4}{3},-\frac{8}{9},\frac{16}{27},\mathrm{...}\right\}$

$a_n=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},\mathrm{...}\right\}$

$a_n=-4\cdot 5^{n-1}$

$a_n=12\cdot {\left(-\frac{1}{2}\right)}^{n-1}$

$a_n=\left\{-2,-4,-8,-16,\mathrm{...}\right\}$

$a_n=\left\{1,3,9,27,\mathrm{...}\right\}$

$a_n=\left\{-4,-12,-36,-108,\mathrm{...}\right\}$

$a_n=\left\{0.8,-4,20,-100,\mathrm{...}\right\}$

$a_n=\{-1.25,-5,-20,-80,\mathrm{...}\}$

$a_n=\left\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125},\mathrm{...}\right\}$

$a_n=\left\{2,\frac{1}{3},\frac{1}{18},\frac{1}{108},\mathrm{...}\right\}$

$a_n=\left\{3,-1,\frac{1}{3},-\frac{1}{9},\mathrm{...}\right\}$

Let $a_1=4,$ $a_n=-3a_{n-1}.$ Find $a_8.$

Let $a_n=-{\left(-\frac{1}{3}\right)}^{n-1}.$ Find $a_{12}.$

$a_n=\left\{-1,3,-9,\mathrm{...},2187\right\}$

$a_n=\left\{2,1,\frac{1}{2},\mathrm{...},\frac{1}{1024}\right\}$

$\begin{array}{cc}{a}_1=1,& r=\frac{1}{2}\end{array}$

$\begin{array}{cc}{a}_1=3,& a_n=2a_{n-1}\end{array}$

$a_n=27\cdot {0.3}^{n-1}$

Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are $200.$

Use explicit formulas to give two examples of geometric sequences whose 7^th terms are $1024.$

Find the 5^th term of the geometric sequence $\{b,4b,16b,\mathrm{...}\}.$

Find the 7^th term of the geometric sequence $\{64a(-b),32a(-3b),16a(-9b),\mathrm{...}\}.$

At which term does the sequence $\{10,12,14.4,17.28,\mathrm{...}\}$ exceed $100?$

At which term does the sequence $\left\{\frac{1}{2187},\frac{1}{729},\frac{1}{243},\frac{1}{81}\mathrm{...}\right\}$ begin to have integer values?

For which term does the geometric sequence $a_{{}_{\mathrm{n}}}=-36{\left(\frac{2}{3}\right)}^{n-1}$ first have a non-integer value?

Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10^th term.

Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8^th term.

Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.