Unit 7: Quadratic Patterns and Combinations of Sequences

A quadratic sequence is a sequence of numbers where the second difference between consecutive terms is constant.

General Formula for a Quadratic Sequence:

\(T_n = an^2 + bn + c\)

Where:

• \(a\), \(b\), and \(c\) are constants determined using the terms of the sequence.
• \(T_n\) is the \(n\)-th term of the sequence.

1. First Differences: The differences between consecutive terms form an arithmetic sequence.
2. Second Differences: The differences between consecutive first differences are constant.

1. Compute the first and second differences: Determine the differences between consecutive terms and then the differences of those differences.
2. Use the relationships:
   • \(2a = \text{second difference}\)
   • \(3a + b = T_2 - T_1\)
   • \(a + b + c = T_1\)
   to solve for \(a\), \(b\), and \(c\).
3. Substitute: Plug the values of \(a\), \(b\), and \(c\) into the general formula \(T_n = an^2 + bn + c\).

Example 1:

Given the sequence: \(6, 12, 22, 36, \dots\)

1. Compute the first and second differences:

\[ \begin{align*} \text{First differences:} & \quad 12 - 6 = 6, \quad 22 - 12 = 10, \quad 36 - 22 = 14 \\ \text{Second differences:} & \quad 10 - 6 = 4, \quad 14 - 10 = 4 \quad (\text{constant}) \end{align*} \]

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

\[ \begin{cases} 2a = 4 \quad \Rightarrow \quad a = 2 \\ 3a + b = 6 \quad \Rightarrow \quad 3(2) + b = 6 \quad \Rightarrow \quad b = 0 \\ a + b + c = 6 \quad \Rightarrow \quad 2 + 0 + c = 6 \quad \Rightarrow \quad c = 4 \end{cases} \] \]

3. General formula:

\(T_n = 2n^2 + 4\)

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Example 2:

Find the 10th term of the sequence \(2n^2 - 3n - 5\):

\[ T_{10} = 2(10)^2 - 3(10) - 5 = 200 - 30 - 5 = 165. \] \]

Sometimes, arithmetic and geometric sequences are combined to create more complex patterns.

Example 3:

Consider a sequence combining an arithmetic (\(a_n = 2n + 1\)) and a geometric (\(g_n = 3^n\)) sequence. The general term is:

\(T_n = a_n + g_n = (2n + 1) + 3^n\)

Find \(T_3\):

\[ T_3 = (2(3) + 1) + 3^3 = 7 + 27 = 34. \] \]

1. Evaluate \(\sum_{k=1}^{5} \left(\frac{3}{4}\right)^k\).
2. If the first term of a geometric series is 200 and the common ratio is 0.75, find the sum to infinity.
3. Determine the sum of the series \(\sum_{n=0}^{\infty} 5 \cdot \left(\frac{1}{3}\right)^n\).
4. The area of each subsequent circle in a sequence is half of the previous one. If the first area is \(50\pi\), find the total area of all circles.

1. Find the sum to infinity of the series \(100, 50, 25, \dots\).

100

150

200

250

2. What is the sum to infinity of the series \(5 + 10 + 20 + \dots\)?

\(\frac{5}{1 - 2}\)

\(\frac{5}{1 - 0.5}\)

\(\frac{5}{1 - 0.75}\)

\(\frac{5}{1 - 0.25}\)

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• Quadratic sequences are characterized by a constant second difference between consecutive terms.
• The general formula for a quadratic sequence is \(T_n = an^2 + bn + c\).
• Combining different types of sequences can create more complex and interesting patterns.
• Understanding these concepts is essential for solving a variety of mathematical and real-world problems.