Question

Solve \(T(n)=T(n-1)+2 n+1\) for \(n \geq 1\) with \(T(0)=1\).

Short answer

The solution to the recurrence is \( T(n) = (n+1)^2 \).

Step-by-step solution

Understand the Recurrence Relation

The given problem is a recurrence relation: \( T(n) = T(n-1) + 2n + 1 \) for \( n \geq 1 \) with an initial condition of \( T(0) = 1 \). This means that each term depends on the previous term plus the function \( 2n + 1 \).

Expand the First Few Terms

Begin expanding terms to look for a pattern: - \( T(1) = T(0) + 2(1) + 1 = 1 + 2 + 1 = 4 \)- \( T(2) = T(1) + 2(2) + 1 = 4 + 4 + 1 = 9 \)- \( T(3) = T(2) + 2(3) + 1 = 9 + 6 + 1 = 16 \)You can see that \( T(n) \) resembles the pattern \( (n+1)^2 \).

Prove the Pattern by Induction

Assume \( T(k) = (k+1)^2 \) holds for some \( k \geq 0 \). **Base Case:** For \( k = 0 \), \( T(0) = 1 \) and \( (0+1)^2 = 1 \); this holds.**Inductive Step:** Assume \( T(k) = (k+1)^2 \), prove \( T(k+1) = (k+2)^2 \).From the recurrence: \[ T(k+1) = T(k) + 2(k+1) + 1 \]Substitute the assumed \( T(k) \): \[ T(k+1) = (k+1)^2 + 2(k+1) + 1 \]\[ T(k+1) = (k+1)(k+1) + 2(k+1) + 1 = (k+2)^2 \]Thus, the formula \( T(n) = (n+1)^2 \) holds.

Conclusion

By expanding and then using mathematical induction, we have proven that the solution to the recurrence relation is \( T(n) = (n+1)^2 \).

Key concepts

Mathematical Induction

Mathematical induction is a method used to prove statements about natural numbers. It's like climbing a ladder. First, you show the base case is true, which is like claiming the bottom rung of the ladder is safe to step on.
Then, you show that if you can safely stand on one rung, you can also step up to the next. This step is known as the inductive step.Let's break it down:

• **Base Case:** This is the starting point. In the exercise, the base case shows that the statement is true for the smallest value of the natural number, which is 0. We verified that for \( k = 0 \), the equation \( T(0) = (0+1)^2 \) holds.
• **Inductive Step:** This is the logical jump from one rung to the next. Assume the statement is true for some arbitrary natural number \( k \). Then prove it for \( k + 1 \). This involves algebraic manipulation using the initial assumption to show that if \( T(k) = (k+1)^2 \), then \( T(k+1) = (k+2)^2 \).

This chain of steps helps confirm that a formula or pattern holds for all natural numbers, provided both steps are successfully proven.

Pattern Recognition

In mathematics, especially in solving recurrence relations, recognizing patterns is a powerful tool. It requires observing the calculated values and identifying a repeating structure or formula. This process often involves working with specific examples and using intuition to see a general rule or sequence emerge from the repetition.The exercise you worked on demonstrates this concept perfectly. As you expand the first few terms of the recurrence relation:

• \( T(1) = 4 \)
• \( T(2) = 9 \)
• \( T(3) = 16 \)

You might observe that these correspond to the perfect squares 2, 3, and 4 squared, respectively. The extracted pattern looks like \( T(n) = (n+1)^2 \).
Pattern recognition is crucial because it allows you to foresee the hidden structure within the recurrence relation, setting up the formulation needed for mathematical induction.
By honing pattern recognition skills, one is better equipped to tackle complex mathematical puzzles where initially formulas aren't apparent.

Discrete Mathematics

Discrete mathematics encompasses a broad area of mathematical concepts, often dealing with countable, distinct elements. Recurrence relations are a key component of discrete mathematics. They provide a way of defining sequences or arrays through recursive formulas, where each term is described using previous terms.In the context of the given exercise, the recurrence relation \( T(n) = T(n-1) + 2n + 1 \) represents a typical problem one might encounter in a discrete math setting.
The goal is often to express these problems in closed-form solutions for easier computation. Solving recurrence relations entails both strategically unraveling the sequence using pattern recognition or by mathematical proofs like induction.
Understanding discrete mathematics is essential for algorithms, computer science, and combinatorics, where solving such relations aids in optimizing and analyzing processes. By learning discrete math, you can develop critical thinking skills to tackle various real-world problems involving discrete structures.