Example 3

The general term can be used to calculate terms in a sequence, or to determine one of the other variables, such as \(t_1 \), \(r \), or \(n \).

Determine \(t_{10} \) of the following geometric sequences.

1. \(t_1 = 0.125\) and \(r = 2\)
   
   
   Solution

   Substitute the given values into the general term formula.
   \(\begin{align}
    t_n &= t_1 r^{n - 1}  \\
    t_{10} &= 0.125\left( 2 \right)^{10 - 1}  \\
    t_{10} &= 64 \\
    \end{align}\)
   
   
   
2. \(t_1 = 100\) and \(r = 0.55\)
   
   
   Solution

   Substitute the given values into the general term formula.
   \(\begin{align}
    t_n &= t_1 r^{n - 1}  \\
    t_{10} &= 100\left( {0.55} \right)^{10 - 1}  \\
    t_{10} &= 0.460\thinspace 536\thinspace 658\thinspace 4... \\
    t_{10} &\buildrel\textstyle.\over =  0.460 \\
    \end{align}\)