13.09 Geometric sequences (Extended)

 

A sequence in which each term increases or decreases from the last by a constant factor is called a geometric sequence. We refer to the constant factor the terms are changing by as the common ratio, which will result from dividing any two successive terms $\left(\frac{u_{n+1}}{u_n}\right)$(un+1​un​​).

We denote the first term in the sequence by the letter $u_1$u1​ , or $a,$a, and the common ratio by the letter $r$r. For example, the sequence $4,8,16,32\dots$4,8,16,32… is geometric with $u_1=4$u1​=4 and $r=2$r=2. The sequence $100,-50,25,-12.5,\dots$100,−50,25,−12.5,… is geometric with $u_1=100$u1​=100 and $r=\frac{-1}{2}$r=−12​.

To describe the rule in words we say "next term is $r$r multiplied by previous term". 

We can find an explicit formula in terms of $u_1$u1​ and $r$r, this is useful for finding the $n$nth term without listing the sequence.

Let's look at a table of a concrete example to see the pattern for the explicit formula. For the sequence $5,10,20,40,\ldots$5,10,20,40,…, we have a starting term of $5$5 and a common ratio of $2$2, that is $u_1=5$u1​=5 and $r=2$r=2. A table of the sequence is show below:

$n$n	$u_n$un​	Pattern
$1$1	$5$5	$5\times2^0$5×20
$2$2	$10$10	$5\times2^1$5×21
$3$3	$20$20	$5\times2^2$5×22
$4$4	$40$40	$5\times2^3$5×23
...
$n$n	$u_n$un​	$5\times2^{n-1}$5×2n−1

 

The pattern starts to become clear and we could guess that the tenth term is $u_{10}=5\times2^9$u10​=5×29  and the one-hundredth term is $u_{100}=5\times2^{99}$u100​=5×299. And following the pattern, the explicit formula for the $n$nth term is $u_n=5\times2^{n-1}$un​=5×2n−1.

For any geometric progression with starting value $u_1$u1​ and common ratio $r$r has the terms given by: $u_1,u_1r,u_1r^2,u_1r^3,...$u1​,u1​r,u1​r2,u1​r3,... We see a similar pattern to our previous table and can write down the formula for the $n$nth term:

$u_n=u_1r^{n-1}$un​=u1​rn−1

 

 

Worked examples

Example 1

For the sequence  $15,30,60,120...$15,30,60,120..., find an explicit rule for the $n$nth term and hence, find the $8$8th term. 

Think: Check that the sequence is geometric, is the next term made by multiplying the previous term by a constant factor? Then write down the starting value $u_1$u1​ and common ratio $r$r and substitute these into the general form: $u_n=u_1r^{n-1}$un​=u1​rn−1

Do: Dividing the second term by the first we get, $\frac{u_2}{u_1}=\frac{30}{15}=\frac{2}{1}$u2​u1​​=3015​=21​. Checking the ratio between the successive pairs we also get $\frac{2}{1}$21​.  So we have a geometric sequence with: $u_1=15$u1​=15 and $r=2$r=2. The general formula for this sequence is: $u_n=15\left(2\right)^{n-1}$un​=15(2)n−1.

Hence, the $8$8th term is:

$u_8$u8​	$=$=	$15\left(2\right)^7$15(2)7
	$=$=	$15\times128$15×128
	$=$=	$1920$1920

 

 

Practice questions

Question 1

Question 2

Question 3