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6.2: Arithmetic and Geometric Sequences

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    Two common types of mathematical sequences are arithmetic sequences and geometric sequences. An arithmetic sequence has a constant difference between each consecutive pair of terms. This is similar to the linear functions that have the form \(y=m x+b .\) A geometric sequence has a constant ratio between each pair of consecutive terms. This would create the effect of a constant multiplier.


    Arithmetic Sequence:
    \(\{5,11,17,23,29,35, \dots\}\)
    Notice here the constant difference is 6. If we wanted to write a general term for this sequence, there are several approaches. One approach is to take the constant difference as the coefficient for the \(n\) term: \(a_{n}=6 n+?\) Then we just need to fill in the question mark with a value that matches the sequence. We could say for the sequence:
    \(\{5,11,17,23,29,35, \dots\}\)
    \(a_{n}=6 n-1\)
    There is also a formula which you can memorize that says that any arithmetic sequence with a constant difference \(d\) is expressed as:
    \(a_{n}=a_{1}+(n-1) d\)
    Notice that if we plug in the values from our example, we get the same answer as before:
    \(a_{n}=a_{1}+(n-1) d\)
    \(a_{1}=5, d=6\)
    So, \(a_{1}+(n-1) d=5+(n-1) * 6=5+6 n-6=6 n-1\)
    or \(a_{n}=6 n-1\)
    If the terms of an arithmetic sequence are getting smaller, then the constant difference is a negative number.
    \(\{24,19,14,9,4,-1,-6, \dots\}\)
    \(a_{n}=-5 n+29\)

    Geometric Sequence
    In a geometric sequence there is always a constant multiplier. If the multiplier is greater than \(1,\) then the terms will get larger. If the multiplier is less than \(1,\) then the terms will get smaller.
    \(\{2,6,18,54,162, \dots\}\)
    Notice in this sequence that there is a constant multiplier of \(3 .\) This means that 3 should be raised to the power of \(n\) in the general expression for the sequence. The fact that these are not multiples of 3 means that we must have a coefficient before the \(3^{n}\)
    \(\{2,6,18,54,162, \dots\}\)
    \(a_{n}=2 * 3^{n-1}\)
    If the terms are getting smaller, then the multiplier would be in the denominator:
    \(\{50,10,2,0.4,0.08, \dots\}\)
    Notice here that each term is begin divided by 5 (or multiplied by \(\frac{1}{5}\) ).
    \(\{50,10,2,0.4,0.08, \ldots .\}\)
    \(a_{n}=\frac{50}{5^{n-1}}\) or \(a_{n}=\frac{250}{5^{n}}\) or \(a_{n}=50 *\left(\frac{1}{5}\right)^{n-1}\) and so on

    Exercises 6.2
    Determine whether each sequence is arithmetic, geometric or neither.
    If it is arithmetic, determine the constant difference.
    If it is geometric determine the constant ratio.
    1) \(\quad\{18,22,26,30,34, \dots\}\)
    2) \(\quad\{9,19,199,1999, \dots\}\)
    3) \(\quad\{8,12,18,27, \dots\}\)
    4) \(\quad\{15,7,-1,-9,-17, \dots\}\)
    5) \(\quad\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots\right\}\)
    6) \(\quad\{100,-50,25,-12.5, \dots\}\)
    7) \(\quad\{-8,12,32,52, \dots\}\)
    8) \(\quad\{1,4,9,16,25, \dots\}\)
    9) \(\quad\{11,101,1001,10001, \ldots\}\)
    10) \(\quad\{12,15,18,21,24, \dots\}\)
    11) \(\quad\{80,20,5,1.25, \dots\}\)
    12) \(\quad\{5,15,45,135,405, \dots\}\)
    13) \(\quad\{1,3,6,10,15, \dots\}\)
    \(\begin{array}{ll}\text { 14) } & \{2,4,6,8,10, \dots\}\end{array}\)
    15) \(\quad\{-1,-2,-4,-8,-16, \dots\}\)
    16) \(\quad\{1,1,2,3,5,8,13,21, \dots\}\)

    This page titled 6.2: Arithmetic and Geometric Sequences is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard W. Beveridge.

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