Let's work through this sequence step by step!

1. First and Second Differences:

The first difference is found by subtracting consecutive terms in the original sequence.

• First difference: $25 - 11$, $45 - 25$, $71 - 45$, $103 - 71$, and $141 - 103$.

The second difference is found by subtracting consecutive terms of the first difference.

• Second difference: Similar steps as above, using the values obtained from the first difference.

Let’s calculate:

Original Sequence: 11, 25, 45, 71, 103, 141

• First difference:

  • $25 - 11 = 14$
  • $45 - 25 = 20$
  • $71 - 45 = 26$
  • $103 - 71 = 32$
  • $141 - 103 = 38$

• Second difference:

  • $20 - 14 = 6$
  • $26 - 20 = 6$
  • $32 - 26 = 6$
  • $38 - 32 = 6$

So, the second difference is constant and equal to 6, which indicates that this sequence is quadratic.

2. Fill in multiples of $n^2$ and subtract:

Here, we compare the original sequence with the sequence of squares of integers:

• $n^2$ values for $n = 1, 2, 3, 4, 5, 6$ are: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, $6^2 = 36$.

Now subtract the multiples of $n^2$ from the original sequence:

• $11 - 1 = 10$
• $25 - 4 = 21$
• $45 - 9 = 36$
• $71 - 16 = 55$
• $103 - 25 = 78$
• $141 - 36 = 105$

3. General formula:

Since the second difference is constant, the sequence is quadratic. The general form of a quadratic sequence is: $a_n = An^2 + Bn + C$ Using the known values from the original sequence and solving for $A$, $B$, and $C$, we can derive the formula.

If we substitute values from the original sequence and solve, we get: $a_n = 3n^2 + 2n + 6$

So, the nth term formula is: $a_n = 3n^2 + 2n + 6$

Do you need any further clarifications? Here are a few related questions you might consider:

1. How are first and second differences used to identify the type of sequence?
2. Can you describe how to derive the general form of a quadratic sequence?
3. Why is the second difference constant for quadratic sequences?
4. What other types of sequences can be identified using difference methods?
5. How would the approach differ for a cubic sequence?

Tip: For quadratic sequences, the second difference is always constant, which helps quickly identify the nature of the sequence.