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  1. Determine the number of terms in the arithmetic sequence \(17, 11, 5, ..., -31\).

    \(\begin{align}
     t_1&= 17 \\
     d&= 11 - 17 = -6 \\
     t_n&= -31 \\
     n&= ? \\
     \end{align}\)
    \(\begin{align}
     t_n&= t_1 + \left( {n - 1} \right)d \\
      -31&= 17 + \left( {n - 1} \right)\left( {-6} \right) \\
      -48&= -6n + 6 \\
      -54&= -6n \\
     9&= n \\
     \end{align}\)
  2. The \(21\)st term of an arithmetic sequence is \(83\).  Determine the first term, given that the common difference is \(3\).

    \(\begin{align}
     t_1&= ? \\
     d&= 3 \\
     t_{21}&= 83 \\
     n&= 21 \\
     \end{align}\)
    \(\begin{align}
     t_n&= t_1 + \left( {n - 1} \right)d \\
     83&= t_1 + \left( {21 - 1} \right)3 \\
     83&= t_1 + 60 \\
     23&= t_1  \\
     \end{align}\)