Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


1 , –5 , 25 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find an equation that describes the sequence \(1, -5, 25, \ldots\), we need to first identify the pattern.

1. **Finding the pattern**:
- The first term \(a_1\) is \(1\).
- The second term \(a_2\) is \(-5\).
- The third term \(a_3\) is \(25\).

2. **Looking at the terms**:
- From \(a_1\) to \(a_2\): \(1 \to -5\) (decrease by \(6\))
- From \(a_2\) to \(a_3\): \(-5 \to 25\) (increase by \(30\))

It appears that the terms are alternating in sign and changing in a non-linear way. A quadratic pattern can often encapsulate such sequences.

3. **Identifying the quadratic function**:
Let's assume the form of the \(n\)th term, \(a_n\), is quadratic:

\[
a_n = An^2 + Bn + C
\]

We can substitute \(n = 1, 2, 3\) to form equations:

- For \(n = 1\) (\(a_1 = 1\)):
\[
A(1)^2 + B(1) + C = 1 \implies A + B + C = 1
\]

- For \(n = 2\) (\(a_2 = -5\)):
\[
A(2)^2 + B(2) + C = -5 \implies 4A + 2B + C = -5
\]

- For \(n = 3\) (\(a_3 = 25\)):
\[
A(3)^2 + B(3) + C = 25 \implies 9A + 3B + C = 25
\]

4. **Setting up the system of equations**:
We now have the following system of equations:
1. \(A + B + C = 1\)
2. \(4A + 2B + C = -5\)
3. \(9A + 3B + C = 25\)

5. **Subtracting equations**:
Now, we'll eliminate \(C\) by subtracting the first equation from the others:

From 2 - 1:
\[
(4A + 2B + C) - (A + B + C) = -5 - 1
\]
\[
3A + B = -6 \quad \text{(Equation 4)}
\]

From 3 - 1:
\[
(9A + 3B + C) - (A + B + C) = 25 - 1
\]
\[
8A + 2B = 24 \quad \text{(Equation 5)}
\]
which simplifies to:
\[
4A + B = 12 \quad \text{(Equation 6)}
\]

6. **Solving the equations**:
Now, we will solve for \(A\) and \(B\) using equations 4 and 6:
From Equation 4:
\[
B = -6 - 3A
\]

Substitute \(B\) into Equation 6:
\[
4A + (-6 - 3A) = 12
\]
\[
4A - 6 - 3A = 12
\]
\[
A - 6 = 12
\]
\[
A = 18
\]

Substituting \(A\) back to find \(B\):
\[
B = -6 - 3(18) = -6 - 54 = -60
\]

Now use \(A\) and \(B\) to find \(C\):
\[
A + B + C = 1 \implies 18 - 60 + C = 1
\]
\[
C = 1 + 60 - 18 = 43
\]

7. **Final equation**:
The final form of the equation for the sequence is:
\[
a_n = 18n^2 - 60n + 43
\]

So, the equation that describes the sequence is:

\[
\boxed{a_n = 18n^2 - 60n + 43}
\]