The sequence below has an $n^{\text {th }}$ term rule of the form $a n^{2}+b n+c$, where $a, b$ and $c$ are whole numbers. Find the $n^{\text {th }}$ term rule for this sequence.
\[
11 \rightarrow 22 \rightarrow 37 \rightarrow 56 \rightarrow 79 \rightarrow
\]
Answer
Final Answer: The $n^{\text{th}}$ term rule for this sequence is \(\boxed{2n^2 + 5n + 4}\)
Step 1 ：Given the sequence: \(11, 22, 37, 56, 79, \dots\)
Step 2 ：We need to find the values of a, b, and c for the quadratic equation \(an^2 + bn + c\).
Step 3 ：Plug in the first few terms of the sequence and solve the system of equations:
Step 4 ：\(a + b + c = 11\)
Step 5 ：\(4a + 2b + c = 22\)
Step 6 ：\(9a + 3b + c = 37\)
Step 7 ：Solve the system of equations to get \(a = 2\), \(b = 5\), and \(c = 4\).
Step 8 ：Final Answer: The $n^{\text{th}}$ term rule for this sequence is \(\boxed{2n^2 + 5n + 4}\)
