For each positive integer n, the quantity s_n is defined such that \(\mathrm{s_{n+2} = (s_n)^2 - s_{n+1}}\). In addition, s_2...

Phase 1: Owning the Dataset

Understanding the Sequence

We have a recursive sequence where:

• \(\mathrm{s_{n+2} = (s_n)^2 - s_{n+1}}\)
• \(\mathrm{s_2 = 1}\) (given)

Let's create a visual showing how the sequence builds:

\(\mathrm{s_1 \rightarrow s_2 \rightarrow s_3 \rightarrow s_4}\)
\(\mathrm{? \quad 1 \quad ? \quad ?}\)

Phase 2: Understanding the Question

Finding the Relationships

Using our recursive formula, let's express \(\mathrm{s_3}\) and \(\mathrm{s_4}\) in terms of \(\mathrm{s_1}\):

For \(\mathrm{s_3}\):

• \(\mathrm{n = 1: s_3 = (s_1)^2 - s_2 = (s_1)^2 - 1}\)

For \(\mathrm{s_4}\):

• \(\mathrm{n = 2: s_4 = (s_2)^2 - s_3 = 1 - s_3}\)
• Substituting \(\mathrm{s_3: s_4 = 1 - ((s_1)^2 - 1) = 2 - (s_1)^2}\)

Key Insight: We need to find values from the answer choices where \(\mathrm{s_4 = 2 - (s_1)^2}\)

Phase 3: Finding the Answer

Systematic Checking

Let's check each possible value of \(\mathrm{s_1}\):

If \(\mathrm{s_1 = -12 \rightarrow s_4 = 2 - (-12)^2 = 2 - 144 = -142}\)
Is -142 in our choices? No, continue.

If \(\mathrm{s_1 = -7 \rightarrow s_4 = 2 - (-7)^2 = 2 - 49 = -47}\)
Is -47 in our choices? No, continue.

If \(\mathrm{s_1 = -3 \rightarrow s_4 = 2 - (-3)^2 = 2 - 9 = -7}\)
Is -7 in our choices? Yes! ✓
Stop here - we found our answer.

Verification

Let's verify our answer works:

• \(\mathrm{s_1 = -3}\)
• \(\mathrm{s_2 = 1}\) (given)
• \(\mathrm{s_3 = (s_1)^2 - s_2 = 9 - 1 = 8}\)
• \(\mathrm{s_4 = (s_2)^2 - s_3 = 1 - 8 = -7}\) ✓

Phase 4: Solution

Final Answer:

• Statement 1 (\(\mathrm{s_1}\)): -3
• Statement 2 (\(\mathrm{s_4}\)): -7

These values satisfy the recursive relationship \(\mathrm{s_{n+2} = (s_n)^2 - s_{n+1}}\) with \(\mathrm{s_2 = 1}\).