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.

-69, 138, -276,
Write your answer using decimals and integers.
an = __ ( __ )^n-1

1 answer

To derive a formula for the sequence -69, 138, -276, we first need to identify a pattern in the sequence.

  1. Identify the first term:

    • The first term (when n = 1) is \( a_1 = -69 \).
  2. Identify the pattern between consecutive terms:

    • The second term (n = 2) is \( a_2 = 138 \).
    • The third term (n = 3) is \( a_3 = -276 \).
    • We can see that the sequence alternates in sign and appears to increase in absolute value.
  3. Calculate the ratios of consecutive terms:

    • \( \frac{a_2}{a_1} = \frac{138}{-69} = -2 \)
    • \( \frac{a_3}{a_2} = \frac{-276}{138} = -2 \)

    This indicates that each term is -2 times the previous term.

  4. Identify the general form of the sequence:

    • The sequence can be defined using an exponential function based on the previous calculations.

The general term can be written as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( r \) is the common ratio.

Substituting the known values:

  • \( a_1 = -69 \)
  • \( r = -2 \)

Therefore, the equation that describes the sequence is: \[ a_n = -69 \cdot (-2)^{(n-1)} \]

So, you can fill in the answer as: \[ a_n = -69 \cdot (-2)^{n-1} \]