Question

find a quadratic model for each sequence.

1. \\(12, 19, 28, 39, 52, 67, \dots\\)
2. \\(-11, -9, -5, 1, 9, 19, \dots\\)
3. \\(8, 3, -6, -19, -36, -57, \dots\\)
4. \\(-7, -2, 9, 26, 49, 78, \dots\\)
5. \\(6, -2, -12, -24, -38, -54, \dots\\)
6. \\(-3, 1, 13, 33, 61, 97, \dots\\)

Explanation:

Find the general quadratic model form

$$ a_n = an^2 + bn + c \quad (n \ge 1) $$

Solve for coefficients of each sequence

For each sequence, use the first three terms to set up a system of linear equations:

$$
LATEXBLOCK0
$$

• Sequence 33: \(12, 19, 28, \dots\)

$$
LATEXBLOCK1
\implies a = 1, \, b = 4, \, c = 7 \implies a_n = n^2 + 4n + 7 $$

• Sequence 34: \(-11, -9, -5, \dots\)

$$
LATEXBLOCK2
\implies a = 1, \, b = -1, \, c = -11 \implies a_n = n^2 - n - 11 $$

• Sequence 35: \(8, 3, -6, \dots\)

$$
LATEXBLOCK3
\implies a = -2, \, b = 1, \, c = 9 \implies a_n = -2n^2 + n + 9 $$

• Sequence 36: \(-7, -2, 9, \dots\)

$$
LATEXBLOCK4
\implies a = 3, \, b = -4, \, c = -6 \implies a_n = 3n^2 - 4n - 6 $$

• Sequence 37: \(6, -2, -12, \dots\)

$$
LATEXBLOCK5
\implies a = -1, \, b = -5, \, c = 12 \implies a_n = -n^2 - 5n + 12 $$

• Sequence 38: \(-3, 1, 13, \dots\)

$$
LATEXBLOCK6
\implies a = 4, \, b = -8, \, c = 1 \implies a_n = 4n^2 - 8n + 1 $$

Verify models with subsequent terms

• Sequence 33: \(a_4 = 16 + 16 + 7 = 39\) (Matches)
• Sequence 34: \(a_4 = 16 - 4 - 11 = 1\) (Matches)
• Sequence 35: \(a_4 = -2(16) + 4 + 9 = -19\) (Matches)
• Sequence 36: \(a_4 = 3(16) - 16 - 6 = 26\) (Matches)
• Sequence 37: \(a_4 = -16 - 20 + 12 = -24\) (Matches)
• Sequence 38: \(a_4 = 4(16) - 32 + 1 = 33\) (Matches)

Answer:

No.	Problem	Answer
34	\(-11, -9, -5, 1, 9, 19, \dots\)	\(a_n = n^2 - n - 11\)
35	\(8, 3, -6, -19, -36, -57, \dots\)	\(a_n = -2n^2 + n + 9\)
36	\(-7, -2, 9, 26, 49, 78, \dots\)	\(a_n = 3n^2 - 4n - 6\)
37	\(6, -2, -12, -24, -38, -54, \dots\)	\(a_n = -n^2 - 5n + 12\)
38	\(-3, 1, 13, 33, 61, 97, \dots\)	\(a_n = 4n^2 - 8n + 1\)