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Determine if the following infinite sequences are arithmetic. If the sequence is arithmetic, determine the common difference, \(d \), and give the next two terms in the sequence.

1. \(101, 93, 86, 80, ... \)
   
   
   Solution

   \(\begin{array}{l}
    d = t_n - t_{n - 1}  \\
    d = 93 - 101 = -8 \\
    d = 86 - 93 = -7 \\
    \end{array}\)
   
   Because the differences of the first two pairs of consecutive terms are not equal, the sequence is not arithmetic.
   
   
   
2. \(21, 85, 149, 213, ... \)
   
   
   Solution

   \(\begin{array}{l}
    d = t_n - t_{n - 1}  \\
    d = 85 - 21 = 64 \\
    d = 149 - 85 = 64 \\
    d = 213 - 149 = 64 \\
    \end{array}\)
   
   The differences are constant; therefore, the sequence is arithmetic. The common difference is 64.
   
   The next two terms are:
   
   \(\begin{align}
    t_5 &= 213 + 64 = 277 \\
    t_6 &= 277 + 64 = 341 \\
    \end{align}\)
   
   
   

Conclusion

The common difference is the first important aspect of an arithmetic sequence; the second is the first term, \(t_1 \). Using these two components, a general term, \(t_n \), can be defined for an arithmetic sequence. The general term can be considered the \(n^{th} \) term of a sequence, or as the formula used to calculate terms of a sequence.