Example 1
Use the graph below to answer the following questions.

  1. List the first five terms of the sequence.

    The first five terms of the sequence are the first five \(y\)-values on the graph.

    In this case, the first five terms are \(750\), \(650\), \(550\), \(450\), and \(350\).
    \(n \)
    		 \(t_n \)
    \(1\) \(750\)
    \(2\) \(650\)
    \(3\) \(550\)
    \(4\) \(450\)
    \(5\) \(350\)

  2. Write the general term of the sequence.

    Use the formula for the general term of arithmetic sequences, noting that \(t_1 = 750 \) and \(d = 650 - 75 = -100 \).

    \(\begin{align}
     t_n &= t_1 + \left( {n - 1} \right)d \\
     t_n &= 750 + \left( {n - 1} \right)\left( { -100} \right) \\
     t_n &= 750 - 100n + 100 \\
     t_n &= 850 - 100n \\
     \end{align} \)
  3. Using the general term, determine \(t_{10}\) and \(t_{15}\).

    Using the formula from above, find \(t_{10} \) and \(t_{15} \).

    \(\begin{align}
     t_n &= 850 - 100n \\
     t_{10} &= 850 - 100\left( {10} \right) \\
     t_{10} &= -150 \\
    \end{align}\)
    \(\begin{align}
     t_n &= 850 - 100n \\
     t_{15} &= 850 - 100\left( {15} \right) \\
     t_{15} &= -650 \\
     \end{align} \)
  4. Determine the slope of the graph.  What connection does the slope have to the general term?

    To calculate slope, use two points on the graph such as \((1, 750)\) and \((4, 450)\), and the slope formula.

    \[\begin{align}
     m &= \frac{{y_2 - y_1 }}{{x_2 - x_1 }} \\
     m &= \frac{{450 - 750}}{{4 - 1}} \\
     m &= \frac{{ -300}}{3} \\
     m &= -100 \\
     \end{align} \]

    What is the connection to the general term? Notice that slope is equal to \(d \)!
  5. Determine the \(y\)-intercept of the graph.  What connection does the \(y\)-intercept have to the general term?

    To find the \(y\)-intercept, extend the "line" towards the \(y\)-axis.  Where the line hits the \(y\)-axis is the \(y\)-intercept.



    According to the graph, the \(y\)-intercept is \(850\). What is the connection to the general term? Notice that \(850\) is the constant term of the formula, \(t_n = 850 - 100n \)!