Glossary Checklist FAQ

C. Arithmetic Sequences and Word Problems

Warm Up

When solving word problems, you can follow these general steps:

1. Read the question carefully, and define the variables based on the problem, including the variable you are trying to solve for.
   
   For example,
   
   \( \begin{align}
    t_1 &= 7 \\
    t_6 &= 77 \\
    n &= 6 \\
    d &= ? \\
    \end{align} \)
   
   Note:  It can be helpful to write out a couple of the terms of the sequence.
   
   
2. Choose the formula(s) you will use to solve the problem.
   
   For example,
   
   \(t_n =  t_1 + (n - 1)d\)
   
   
3. Substitute known values for the variables.
   
   For example,
   
   \(77 = 7 + (6 - 1)d \)
   
   
4. Solve the problem, and check your answer by putting it back into the original equation to ensure that it makes sense. It is also important to consider whether the solution makes sense in the context of the problem.
   
   For example,
   
   \( \begin{align}
    77 &= 7 + \left( {6 - 1} \right)d \\
    70 &= 5d \\
    14 &= d \\
    \end{align} \)
   
   Check:
   
   \( \begin{align}
    t_n &= t_1  + \left( {n - 1} \right)d \\
    77 &= 7 + \left( {6 - 1} \right)14 \\
    77 &= 7 + 5\left( {14} \right) \\
    77 &= 7 + 70 \\
    77 &= 77 \\ 
    \end{align} \)
   
   The answer is verified. The common difference is \(14\).