Write an equation to describe the sequence below.Use n to represent the position of a term in the sequence,where n = 1 for the first term.

-99, 198/5 , -396/25 ,
Write your answer using proper fractions, improper fractions, and integers.
an = __ ( __ )^n-1

1 answer

To find a formula for the given sequence \(-99, \frac{198}{5}, -\frac{396}{25}\), we will express the \(n\)-th term of the sequence \(a_n\) in the form \(a_n = a \cdot r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

  1. First Term \(a\):

    • The first term \(a_1\) is \(-99\). Therefore, \(a = -99\).
  2. Finding the Common Ratio \(r\):

    • To find \(r\), we divide the second term by the first term: \[ r = \frac{a_2}{a_1} = \frac{\frac{198}{5}}{-99} \] To simplify this, we can multiply the numerator and denominator: \[ r = \frac{198}{5} \cdot \frac{-1}{99} = \frac{198 \cdot -1}{99 \cdot 5} = \frac{-198}{495} \] Simplifying \(\frac{-198}{495}\):
    • \(198\) and \(495\) can both be divided by \(99\): \[ r = \frac{-2}{5} \]
  3. Formulating the nth Term:

    • We have \(a = -99\) and \(r = -\frac{2}{5}\). Thus, the formula for \(a_n\) becomes: \[ a_n = -99 \cdot \left(-\frac{2}{5}\right)^{n-1} \]
  4. Final Expression:

    • Therefore, the equation that describes the sequence is: \[ a_n = -99 \left(-\frac{2}{5}\right)^{n-1} \]

Final answer: \[ a_n = -99 \left(-\frac{2}{5}\right)^{n-1} \]