To determine the quadratic function $f(n) = an^2 + bn + c$ that describes the given sequence, we can follow these steps:

Step 1: Identify Patterns

Let's look at the given terms of the sequence:

• $f(1) = 2$
• $f(2) = 6$
• $f(3) = 12$
• $f(4) = 20$
• $f(5) = 30$

Step 2: Set Up Equations for Known Points

Since we know that $f(n) = an^2 + bn + c$, we can substitute each term to create equations:

1. For $n = 1$:
   $a(1)^2 + b(1) + c = 2 \quad \Rightarrow \quad a + b + c = 2$

2. For $n = 2$:
   $a(2)^2 + b(2) + c = 6 \quad \Rightarrow \quad 4a + 2b + c = 6$

3. For $n = 3$:
   $a(3)^2 + b(3) + c = 12 \quad \Rightarrow \quad 9a + 3b + c = 12$

Step 3: Solve the System of Equations

Now we have a system of three equations:

1. $a + b + c = 2$
2. $4a + 2b + c = 6$
3. $9a + 3b + c = 12$

Subtract equation (1) from equation (2):

$(4a + 2b + c) - (a + b + c) = 6 - 2$ $3a + b = 4$

Now subtract equation (2) from equation (3):

$(9a + 3b + c) - (4a + 2b + c) = 12 - 6$ $5a + b = 6$

Now we have two equations with two variables:

1. $3a + b = 4$
2. $5a + b = 6$

Subtract the first equation from the second:

$(5a + b) - (3a + b) = 6 - 4$ $2a = 2 \quad \Rightarrow \quad a = 1$

Substitute $a = 1$ into $3a + b = 4$:

$3(1) + b = 4 \quad \Rightarrow \quad b = 1$

Finally, substitute $a = 1$ and $b = 1$ into $a + b + c = 2$:

$1 + 1 + c = 2 \quad \Rightarrow \quad c = 0$

Step 4: Write the Quadratic Function

The quadratic function that describes the sequence is:

$f(n) = n^2 + n$

Verification

Let's verify by plugging in the values of $n$:

• $f(1) = 1^2 + 1 = 2$
• $f(2) = 2^2 + 2 = 6$
• $f(3) = 3^2 + 3 = 12$
• $f(4) = 4^2 + 4 = 20$
• $f(5) = 5^2 + 5 = 30$

This matches the sequence, confirming that the function is correct.

Would you like further details on any part of this solution?

Here are some related questions you might find helpful:

1. How do you determine if a sequence can be represented by a quadratic function?
2. What are the steps to solve a system of equations for quadratic sequences?
3. How can you identify the pattern in sequences that aren't quadratic?
4. Can this method be used for higher-order polynomial sequences?
5. How does changing coefficients in a quadratic function affect its shape?

Tip: For sequences, check the first and second differences of terms to identify if it's quadratic (constant second difference).