Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


–1 , –3 , –9 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

The sequence given is:

-1, -3, -9, ...

To find a pattern, let's observe the relationship between the terms.

1. The first term \( a_1 = -1 \).
2. The second term \( a_2 = -3 \).
3. The third term \( a_3 = -9 \).

We can see that the terms are decreasing in a specific manner. Let's analyze the possibility of relating these terms to powers or multiplication.

If we look closely:
- The second term \(-3\) can be expressed as \(-1 \times 3\).
- The third term \(-9\) can be expressed as \(-3 \times 3\).

This suggests that each term can be derived from the previous term by multiplying it by \(3\).

To formalize this observation, we can define the \(n\)-th term of the sequence using a recursive approach; however, since we need a direct formula, let's analyze:

The terms can also be written:
- \( a_1 = -1 = -1 \times 3^0 \)
- \( a_2 = -3 = -1 \times 3^1 \)
- \( a_3 = -9 = -1 \times 3^2 \)

From this, we can deduce that the \(n\)-th term can be expressed as:
\[
a_n = -1 \times 3^{n-1}
\]
This can be simplified to:
\[
a_n = -3^{n-1}
\]

Thus, the equation you requested, where \(n\) represents the position of a term (with \(n=1\) for the first term), is:

\[
a_n = -3^{n-1}
\]