finding the sequence of general terms of a sequence 1,6,13,22,33

To find the general term (or \( n \)th term) of the sequence \( 1, 6, 13, 22, 33 \), we need to analyze the pattern in the sequence.

### Step 1: Analyze the sequence
Let's look at the differences between consecutive terms:

\[
6 - 1 = 5 \\
13 - 6 = 7 \\
22 - 13 = 9 \\
33 - 22 = 11
\]

The first differences are: \( 5, 7, 9, 11 \).

### Step 2: Analyze the second differences
Now, let's look at the differences between the first differences:

\[
7 - 5 = 2 \\
9 - 7 = 2 \\
11 - 9 = 2
\]

The second differences are constant, equal to \( 2 \). This indicates that the sequence is quadratic, i.e., the general term \( a_n \) will be of the form:

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

### Step 3: Determine coefficients \( a \), \( b \), and \( c \)
We now have to find the coefficients \( a \), \( b \), and \( c \) using the first few terms of the sequence.

1. For \( n = 1 \): \( a_1 = a(1)^2 + b(1) + c = a + b + c = 1 \)
2. For \( n = 2 \): \( a_2 = a(2)^2 + b(2) + c = 4a + 2b + c = 6 \)
3. For \( n = 3 \): \( a_3 = a(3)^2 + b(3) + c = 9a + 3b + c = 13 \)

This gives us the following system of equations:

\[
a + b + c = 1 \quad \text{(Equation 1)}
\]
\[
4a + 2b + c = 6 \quad \text{(Equation 2)}
\]
\[
9a + 3b + c = 13 \quad \text{(Equation 3)}
\]

### Step 4: Solve the system of equations

Subtract Equation 1 from Equation 2:

\[
(4a + 2b + c) - (a + b + c) = 6 - 1 \\
3a + b = 5 \quad \text{(Equation 4)}
\]

Subtract Equation 2 from Equation 3:

\[
(9a + 3b + c) - (4a + 2b + c) = 13 - 6 \\
5a + b = 7 \quad \text{(Equation 5)}
\]

Now, subtract Equation 4 from Equation 5:

\[
(5a + b) - (3a + b) = 7 - 5 \\
2a = 2 \quad \Rightarrow a = 1
\]

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

\[
3(1) + b = 5 \\
b = 2
\]

Substitute \( a = 1 \) and \( b = 2 \) into Equation 1:

\[
1 + 2 + c = 1 \\
c = -2
\]

### Step 5: Write the general term

Thus, the general term of the sequence is:

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

This formula can be used to find any term in the sequence.

Would you like details on how to verify this result or any further questions?

### Related Questions:
1. What is the next term in the sequence using the general formula?
2. How would the sequence change if the constant term \( c \) was different?
3. Can you derive the general term of a cubic sequence?
4. What is the sum of the first \( n \) terms of the sequence?
5. How would you identify a sequence that is neither arithmetic nor geometric?
6. What are the first 10 terms of the sequence based on the general term?
7. How does this quadratic sequence compare to a simple arithmetic sequence?
8. How can you use the general term to solve real-world problems?

### Tip:
When identifying patterns in sequences, always check the differences between terms, as constant differences indicate arithmetic sequences, and constant second differences indicate quadratic sequences.

Discover how to find the general term of the sequence 1, 6, 13, 22, 33 using a step-by-step quadratic equation approach. This problem involves analyzing patterns in sequences and applying quadratic formula techniques. Suitable for advanced high school students.

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