Sequences and Series

Hong Kong

Stage 4 - Stage 5

Lesson

Sometimes we are asked to find the common ratio of a certain geometric progression, and then use this to find a group of terms or simply a specific term. We need to remember that the $n$`n`th term of a GP is given by:

$t_n=ar^{n-1}$`t``n`=`a``r``n`−1

Note that there are two variables in the formula – the first term $a$`a` and the common ratio $r$`r` and any set of three consecutive terms will allow us to evaluate $r$`r`.

Take for example the geometric progression beginning $3,12,48,...$3,12,48,... . The common ratio is simply the ratio $\frac{t_2}{t_1}=\frac{t_3}{t_2}=4$`t`2`t`1=`t`3`t`2=4.

This means that we can write down a formula for the $n$`n`th term as

$t_n$tn |
$=$= | $ar^{n-1}$arn−1 |

$=$= | $3\left(4\right)^{n-1}$3(4)n−1 |

and this in turn allows us to determine any term or sequence of terms we like. For example, we see that the $5$5th term is given by $t_5=3\times4^4=768$`t`5=3×44=768.

As another example, we might wonder whether the sequence $\sqrt{3},6,12\sqrt{3}$√3,6,12√3 is geometric. It may not be immediately obvious, but we can show that the numbers are in geometric sequence in two ways. In the first method, we note that both

$\frac{6}{\sqrt{3}}$6√3 | $=$= | $\frac{6}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}$6√3×√3√3 |

$=$= | $2\sqrt{3}$2√3 |

and that $\frac{12\sqrt{3}}{6}=2\sqrt{3}$12√36=2√3, and so it is immediately geometric with $r=2\sqrt{3}$`r`=2√3.

In the second method we note that in any geometric progression, $\frac{t_2}{t_1}=\frac{t_3}{t_2}$`t`2`t`1=`t`3`t`2 and by "cross" multiplication, we see that $\left(t_2\right)^2=t_1\times t_3$(`t`2)2=`t`1×`t`3 or that the middle term is always the square root of the product of terms on each side of it. The middle term is defined to be the geometric mean of the outer two terms so that $t_2=\sqrt{t_1\times t_3}$`t`2=√`t`1×`t`3 .

In a final example consider the three terms $2,x,32$2,`x`,32 where the middle term is unknown. If this sequence is geometric then we can find $x$`x`, for we know that $\frac{x}{2}=\frac{32}{x}$`x`2=32`x` and so $x^2=64$`x`2=64 and this means that there are two possible solutions for $x$`x` as $x=8$`x`=8 or $x=-8$`x`=−8. The two sequences are thus $2,8,32$2,8,32 or $2,-8,32$2,−8,32.

Study the pattern for the following sequence.

$-9$−9$,$, $3.6$3.6$,$, $-1.44$−1.44$,$, $0.576$0.576 ...

State the common ratio between the terms.

Study the pattern for the following geometric sequence, and write down the next two terms.

$3$3, $15$15, $75$75, $\editable{}$, $\editable{}$

Study the pattern for the following geometric sequence, and write down the next two terms.

$12$12, $-48$−48, $192$192, $\editable{}$, $\editable{}$