find the nth term for the quadratic sequence -2,7,22,43,70,103

The given sequence is:  
\[-2, 7, 22, 43, 70, 103\]

This sequence appears to be quadratic, so we will find the \(n\)th term using the general quadratic form:
\[
T_n = an^2 + bn + c
\]
Where:
- \(a\), \(b\), and \(c\) are constants that we will determine.

### Step 1: Find the first and second differences.

- First differences:
  \[
  7 - (-2) = 9,\quad 22 - 7 = 15,\quad 43 - 22 = 21,\quad 70 - 43 = 27,\quad 103 - 70 = 33
  \]
  So the first differences are: \(9, 15, 21, 27, 33\)

- Second differences:
  \[
  15 - 9 = 6,\quad 21 - 15 = 6,\quad 27 - 21 = 6,\quad 33 - 27 = 6
  \]
  The second differences are constant at 6, confirming that this is a quadratic sequence.

### Step 2: Solve for \(a\), \(b\), and \(c\).

For a quadratic sequence, the second difference is equal to \(2a\). So:
\[
2a = 6 \implies a = 3
\]

Now, using the general form \(T_n = 3n^2 + bn + c\), substitute the first few terms of the sequence to find \(b\) and \(c\).

- For \(n = 1\):
  \[
  T_1 = 3(1)^2 + b(1) + c = -2
  \implies 3 + b + c = -2 \implies b + c = -5 \quad (1)
  \]

- For \(n = 2\):
  \[
  T_2 = 3(2)^2 + b(2) + c = 7
  \implies 12 + 2b + c = 7 \implies 2b + c = -5 \quad (2)
  \]

### Step 3: Solve the system of equations.

From equation (1):
\[
b + c = -5
\]

From equation (2):
\[
2b + c = 7
\]

Subtract equation (1) from equation (2):
\[
(2b + c) - (b + c) = 7 - (-5)
\implies b = 12
\]

Now, substitute \(b = 12\) into equation (1):
\[
12 + c = -5 \implies c = -17
\]

### Step 4: Write the nth term.

Now that we have \(a = 3\), \(b = 12\), and \(c = -17\), the \(n\)th term is:
\[
T_n = 3n^2 + 12n - 17
\]

### Final Answer:
The nth term for the given quadratic sequence is:
\[
T_n = 3n^2 + 12n - 17
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How can you use the first and second differences to identify quadratic sequences?
2. How would the formula change if the second differences were not constant?
3. Can you apply this method to a cubic sequence?
4. What if the initial term was a different number? How would that affect the sequence?
5. Can this method be used for sequences with negative second differences?

**Tip:** When identifying quadratic sequences, always check for constant second differences to confirm the pattern.

Find the nth term for the quadratic sequence -2, 7, 22, 43, 70, 103.

Learn how to find the nth term of the quadratic sequence -2, 7, 22, 43, 70, 103 with a detailed step-by-step solution. This tutorial covers quadratic sequences, the general form T_n = an^2 + bn + c, and solving for coefficients using differences. Suitable for students in Grades 9-12.

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