Write down an expression for this sequence: -1, 3, 9, 17, 27, 39

To find an expression for the sequence \(-1, 3, 9, 17, 27, 39\), let's analyze the pattern:

### Step 1: Identify the sequence pattern
The sequence appears to be neither purely arithmetic nor geometric, so let's look at the differences between the terms:

- \(3 - (-1) = 4\)
- \(9 - 3 = 6\)
- \(17 - 9 = 8\)
- \(27 - 17 = 10\)
- \(39 - 27 = 12\)

The differences between consecutive terms are \(4, 6, 8, 10, 12\), which form an arithmetic sequence with a common difference of \(2\).

### Step 2: Formulate the general term expression
Given that the first differences themselves form an arithmetic sequence, we can express this sequence as a quadratic sequence of the form:

\[
a_n = an^2 + bn + c
\]

where \(n\) represents the position of the term in the sequence.

### Step 3: Set up equations to solve for \(a\), \(b\), and \(c\)
Using the first three terms, we set up equations:

For \(n = 1\), \(a_1 = -1\):
\[
a(1)^2 + b(1) + c = -1 \quad \text{(Equation 1)}
\]

For \(n = 2\), \(a_2 = 3\):
\[
a(2)^2 + b(2) + c = 3 \quad \text{(Equation 2)}
\]

For \(n = 3\), \(a_3 = 9\):
\[
a(3)^2 + b(3) + c = 9 \quad \text{(Equation 3)}
\]

Simplifying the equations:

1. From Equation 1: \(a + b + c = -1\)
2. From Equation 2: \(4a + 2b + c = 3\)
3. From Equation 3: \(9a + 3b + c = 9\)

### Step 4: Solve the system of equations

Subtract Equation 1 from Equation 2:

\[
(4a + 2b + c) - (a + b + c) = 3 - (-1)
\]

\[
3a + b = 4 \quad \text{(Equation 4)}
\]

Subtract Equation 2 from Equation 3:

\[
(9a + 3b + c) - (4a + 2b + c) = 9 - 3
\]

\[
5a + b = 6 \quad \text{(Equation 5)}
\]

Subtract Equation 4 from Equation 5:

\[
(5a + b) - (3a + b) = 6 - 4
\]

\[
2a = 2 \implies a = 1
\]

Substituting \(a = 1\) into Equation 4:

\[
3(1) + b = 4 \implies b = 1
\]

Substituting \(a = 1\) and \(b = 1\) into Equation 1:

\[
1 + 1 + c = -1 \implies c = -3
\]

### Step 5: General term formula

Thus, the expression for the \(n\)-th term of the sequence is:

\[
a_n = n^2 + n - 3
\]

Would you like more details or have any specific questions? Here are some related questions to expand on this topic:

1. How can you verify if a sequence is quadratic?
2. What are some other methods to find the general term of a sequence?
3. How does this quadratic sequence compare to an arithmetic or geometric sequence?
4. Can you find the sum of the first \(n\) terms of this sequence?
5. How do the first differences of a quadratic sequence form an arithmetic sequence?

**Tip:** When dealing with sequences that are neither arithmetic nor geometric, check if the differences between terms form a recognizable pattern, such as a second sequence being arithmetic, which indicates a quadratic relationship!

Learn how to find the general term expression for the sequence -1, 3, 9, 17, 27, 39 with a detailed step-by-step solution. This problem covers quadratic sequences and their general term formula. Suitable for high school students.

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