Question

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

Short answer

The solution is \( T(n) = 1 + 2n \).

Step-by-step solution

Understanding the Problem

We are given a recursive relation: \( T(n) = T(n-1) + 2 \) with the initial condition \( T(0) = 1 \). Our task is to find a formula for \( T(n) \) in terms of \( n \).

Identify Recurrence Pattern

To find the pattern, substitute small values of \( n \). Begin with the initial condition: \( T(0) = 1 \). Then, using the recurrence relation, calculate subsequent terms: \( T(1) = T(0) + 2 = 1 + 2 = 3 \), \( T(2) = T(1) + 2 = 3 + 2 = 5 \), \( T(3) = T(2) + 2 = 5 + 2 = 7 \), and so on. We notice a pattern: \( T(n) = 1 + 2n \).

Formulating the General Solution

Observing the results, it shows that each term increases by 2 on the previous term, starting from 1. Thus, a general formula can be formulated as \( T(n) = 1 + 2n \).

Verification of the Solution

Verify the formula by plugging different values of \( n \) into it. For \( n = 0 \), \( T(0) = 1 + 2 \times 0 = 1 \), which matches the initial condition. For \( n = 1 \), \( T(1) = 1 + 2 \times 1 = 3 \), and for \( n = 2, \) \( T(2) = 1 + 2 \times 2 = 5 \). The pattern holds true.

Key concepts

Recursive Formula

In mathematics, a recursive formula can be a handy tool for generating the terms of a sequence. It provides a rule or step-by-step process by which we can determine any term in the sequence based on the preceding terms. For example, in the problem we are dealing with, the recursive formula is given by:

• \( T(n) = T(n-1) + 2 \)

This specific formula tells us how to calculate the term \( T(n) \) by adding 2 to the previous term, \( T(n-1) \). The beauty of such formulas is that they allow complex sequences to be broken down into simpler operations, often involving just basic arithmetic.

Recursive formulas are more than just mathematical tools; they are integral to understanding and expressing patterns, often providing a step-by-step guide that highlights the iterative nature of sequence building. It's like using a recipe that tells you how to prepare the next step based on what you've already done. They are widely used in computer science for algorithms and in mathematical problem solving where direct computation of terms is not feasible or efficient.

Initial Conditions

Initial conditions act as the starting point or base case for a recursive formula. They are critical because, without them, we wouldn't be able to determine the first term needed to apply the recursive formula. A recursive formula relies entirely on knowing an initial or previous term, and without an initial condition, the sequence would be undefined.For example, in our problem, the initial condition is \( T(0) = 1 \). This tells us that when \( n \) is 0, \( T(n) \) is equal to 1. From here, we can use the recursive formula to derive subsequent terms. Initial conditions help ensure that even complex sequences are anchored to a real, tangible starting point, making the progression clearer.

These conditions are vital to confirming all derived terms correctly represent the sequence as intended by its rule. Additionally, for problems or sequences involving real-world scenarios, the initial condition often aligns with a known fact or measurement, giving further practical importance to its role.

Pattern Recognition

Pattern recognition is an essential skill when working with sequences and their recursive relations. By carefully observing computed terms, one can often see a predictable and systematic development that suggests a broader formula.In our example problem, by calculating multiple terms using the recursive relationship, we identified a clear pattern. The sequence we constructed—from \( T(0) \) onwards—is:

• \( T(0) = 1 \)
• \( T(1) = 3 \)
• \( T(2) = 5 \)
• \( T(3) = 7 \)

Upon examining these terms, a pattern emerges: the terms increase consistently by 2, forming the general expression \( T(n) = 1 + 2n \). This pattern recognition allows us to leap from understanding individual terms to grasping the entire sequence's nature.Recognizing patterns helps not only in simplifying calculations but also in verifying whether results align with expected outcomes. It's a fundamental aspect of sequence analysis, ensuring consistency and correctness in mathematical reasoning.