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      <text>16.1.2.1 Arithmetic and geometric progressions</text>
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      <text><![CDATA[<h3>Arithmetic and geometric progressions</h3><br>
<p>Consider the sequences: \[ 1,\ 4,\ 7,\ 10,\ \ldots \quad \text{and} \quad 3,\ 1,\ -1,\ -3,\ \ldots\]In both, any particular term is obtained from the previous term by the <strong>addition</strong> of a constant value (\(3\) and \(-2\) respectively). Each of these sequences are said to be an <strong>arithmetic sequence</strong> or <strong>arithmetic progression</strong> and has general form: \[ a,\ a+d,\ a+2d,\ a+3d,\ \ldots ,\ a+(n-1)d,\ \ldots\] in which \(a,d\) are given numbers. In the first example above \(a=1,\ d=3\) whereas, in the second example, \(a=3,\ d=-2\). The difference between any two successive terms of a given arithmetic sequence gives the value of \(d\) which is called the <strong>common difference</strong>.</p> 
<p>Two sequences which are <strong>not</strong> arithmetic sequences are: \[ 1,\ 2,\ 4,\ 8,\ \ldots\] \[ -1,\ -\dfrac{1}{3},\ -\dfrac{1}{9},\ -\dfrac{1}{27},\ \ldots\]In each case a particular term is obtained from the previous term by <strong>multiplying</strong> by a constant factor (\(2\) and \(\dfrac{1}{3}\) respectively). Each is an example of a <strong>geometric sequence</strong> or <strong>geometric progression</strong> with the general form: \[ a,\ ar,\ ar^2,\ ar^3,\ \ldots\]where ‘\(a\)’ is the first term and \(r\) is called the <strong>common ratio</strong>, being the ratio of two successive terms. In the first geometric sequence above \(a=1\), \(r=2\) and in the second geometric sequence \(a=-1\), \(r=\dfrac{1}{3}\).</p>]]></text>
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