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A closed formula is a way to express the terms of a sequence using a straightforward, non-recursive function. For sequences defined by recurrence relations, a closed formula offers a direct method to compute any term without computing all preceding terms.
In the given problem, the sequence starts with terms 2, 5, 13, 35, and is defined by the recurrence relation:

• \( u_{n+2} = 5u_{n+1} - 6u_{n} \)

To derive the closed formula for this sequence, we solve the associated characteristic equation. Once the roots of this characteristic equation are known, they help to build a solution of the form:

• \( u_n = A \cdot x_1^n + B \cdot x_2^n \)

Where \(x_1\) and \(x_2\) are the roots. The constants \(A\) and \(B\) are determined using the initial conditions of the sequence. For our solution:

• \( x_1 = 2 \)
• \( x_2 = 3 \)

Thus, the closed formula becomes \( u_n = 2^n + 3^n \), which directly calculates the \(n^{th}\) term of the sequence.

Mathematical induction is a powerful proof technique used to establish the truth of an infinite sequence of statements. In this context, induction is used to verify that the closed formula \( u_n = 2^n + 3^n \) works for all terms of the sequence defined by the recurrence relation.

• Base Case: Start by verifying the formula for initial values. For our sequence, we check \( n = 0 \) and \( n = 1 \) which confirm that \( u_0 = 2 \) and \( u_1 = 5 \) both hold true in the formula.
• Inductive Step: Assume the formula holds for \( n = k \), meaning \( u_k = 2^k + 3^k \) is correct. Then prove it for \( n = k+1 \). By plugging into the recurrence:
  • \( u_{k+2} = 5u_{k+1} - 6u_k \)
  We substitute using the inductive hypothesis and simplify to show:
  • \( u_{k+2} = 2^{k+2} + 3^{k+2} \)
  Thus, the formula stands true for all \( n \geq 0 \).

Mathematical induction confirms that our closed formula correctly represents the sequence across all indices.

The characteristic equation is an integral aspect of solving linear recurrence relations. By converting the recurrence into a polynomial equation, we can find the roots, which represent possible forms of solutions.

• From Recurrence Relation: For any sequence \( u_n \), the recurrence relation is given by \( u_{n+2} = 5u_{n+1} - 6u_n \).
  Create the characteristic equation \( x^2 = 5x - 6 \), which simplifies to:
• \( x^2 - 5x + 6 = 0 \)

• Solving the Quadratic: Factor the quadratic to find the roots:
• \( (x-2)(x-3) = 0 \)
• This provides roots \( x = 2 \) and \( x = 3 \).

• Application of Roots: Use these roots to define a general solution for \( u_n \):
• \( u_n = A \cdot 2^n + B \cdot 3^n \)

The characteristic equation thus helps obtain the structure of the sequence's closed formula, ensuring that the sequence adheres to its initial defining properties.