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!