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      <text>8.1.11 (Geometric sequences)</text>
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      <text><![CDATA[<h1>Geometric sequences</h1>

<p>So far, we have discussed arithmetic sequences, that is sequences whose difference between terms is constant. These are not the only form of sequence. In this section we introduce and examine geometric sequences.</p>

<p>A <b>geometric sequence</b> is one in which the ratio of consecutive terms is constant. For example,
\[ 
1, 2, 4, 8, 16, \ldots 
\]
is a geometric sequence since each term is twice as big as the previous one. We can check the ratios:
\[ 
\frac{2}{1}=2,\quad \frac{4}{2}=2,\quad \frac{8}{4}=2,\quad \text{etc.} 
\]
So in a geometric sequence, we find that each term is obtained from the previous one by multiplying by a fixed number, which we call the <b>common ratio</b> of the sequence. This is usually denoted \(r\) -- so for the example above, \(r=2\).</p>
<>br/
<p>We can use a recurrence relation to capture the idea of multiplying each term by the common ratio to get the next term.For example, the sequence \(1,2,4,8,\ldots\) is given by the recurrence relation
\[ 
u_{n+1}=2u_n,\quad u_1=1. 
\]
In general, if the first term is \(a\) and the common ratio is \(r\), the recurrence relation is
\[ 
u_{n+1}=ru_n,\quad u_1=a. 
\]</p>
<br/>
<p>We can also define such a sequence by a formula. The sequence with first term \(a\) and common ratio \(r\) has the following terms:
\[ 
a, ar, ar^2, ar^3,\ldots. 
\]
We can see from this that the term \(u_n\) is given by the formula
\[ 
u_n=a r^{n-1}, 
\]
where the first term is \(u_1\).</p>]]></text>
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