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.

-1, 4, -16, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

1 answer

To find the equation for the sequence \(-1, 4, -16, \ldots\), we first observe the pattern of the terms:

  • The first term \(a_1 = -1\)
  • The second term \(a_2 = 4\)
  • The third term \(a_3 = -16\)

Looking closely at the values of the terms, we note the following:

  1. The signs alternate, with the first term being negative, the second positive, and the third negative.
  2. The absolute values of the terms are \(1, 4, 16\), which can be expressed in terms of powers of \(4\):
    • \(|a_1| = 1 = 4^0\)
    • \(|a_2| = 4 = 4^1\)
    • \(|a_3| = 16 = 4^2\)

This suggests that the magnitude of the \(n\)-th term is \(4^{n-1}\).

To capture the alternating signs, we can use the term \((-1)^n\), which will let the terms alternate between negative and positive as follows:

  • For \(n=1\): \((-1)^1 = -1\)
  • For \(n=2\): \((-1)^2 = 1\)
  • For \(n=3\): \((-1)^3 = -1\)

Putting this together, the general term \(a_n\) can be expressed as: \[ a_n = (-1)^n \cdot 4^{n-1} \]

Thus the equation for the sequence is:

\[ a_n = (-1)^n (4)^{n-1} \]