Chapter 9: Problem 5
Solve \(T(n)=T(n-1)+2\) for \(n \geq 1\) with \(T(0)=2\).
Short Answer
Expert verified
The solution is \( T(n) = 2 + 2n \) for all integer \( n \geq 0 \).
Step by step solution
01
Understand the Problem
We are given a recursive relation \( T(n) = T(n-1) + 2 \) with the initial condition \( T(0) = 2 \). Our task is to find an explicit formula for \( T(n) \) in terms of \( n \).
02
Identify the Pattern
To find a pattern, let's compute the first few values of \( T(n) \) using the recursive relation:- \( T(0) = 2 \)- \( T(1) = T(0) + 2 = 2 + 2 = 4 \)- \( T(2) = T(1) + 2 = 4 + 2 = 6 \)- \( T(3) = T(2) + 2 = 6 + 2 = 8 \)It seems that \( T(n) \) increases by 2 for each increment of \( n \). This suggests a linear sequence.
03
Derive the General Formula
Given the pattern observed in the previous step, we hypothesize that \( T(n) \) is a linear function of \( n \) of the form \( T(n) = an + b \). From the values computed, we observe that \( T(n) = 2 + 2n \). We confirm this by checking if it satisfies the recursive relation and initial condition.
04
Verify the Formula
Check if \( T(n) = 2 + 2n \) satisfies the recursive equation:- For the initial condition: \( T(0) = 2 \), our formula gives \( 2 + 2 \times 0 = 2 \).- Assuming \( T(k) = 2 + 2k \), then \( T(k+1) = T(k) + 2 = (2 + 2k) + 2 = 2 + 2k + 2 = 2 + 2(k+1) \).This confirms that \( T(n) = 2 + 2n \) is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Explicit Formula
An explicit formula allows us to determine any term in a sequence directly, without needing the previous terms. In the context of our exercise, we are tasked with finding an explicit formula for the recursive relation given by \( T(n) = T(n-1) + 2 \) with the initial condition \( T(0) = 2 \).
By examining the first few terms of the sequence—\( T(0) = 2 \), \( T(1) = 4 \), \( T(2) = 6 \), and so on—we find that each subsequent term increases by 2. Our goal is to express this pattern in a formula that directly computes \( T(n) \) for any \( n \).
Observing the sequence progression, we can hypothesize that \( T(n) = 2 + 2n \). The explicit formula lets us bypass the recursive steps and directly calculate, for example, \( T(5) = 2 + 2\times5 = 12 \). Using an explicit formula is especially beneficial for computing distant terms in large sequences quickly and efficiently.
By examining the first few terms of the sequence—\( T(0) = 2 \), \( T(1) = 4 \), \( T(2) = 6 \), and so on—we find that each subsequent term increases by 2. Our goal is to express this pattern in a formula that directly computes \( T(n) \) for any \( n \).
Observing the sequence progression, we can hypothesize that \( T(n) = 2 + 2n \). The explicit formula lets us bypass the recursive steps and directly calculate, for example, \( T(5) = 2 + 2\times5 = 12 \). Using an explicit formula is especially beneficial for computing distant terms in large sequences quickly and efficiently.
Linear Sequence
A linear sequence is a type of arithmetic sequence where each term is derived from adding a constant value to the previous term. In our example, \( T(n) = T(n-1) + 2 \), the formula signifies that the difference between consecutive terms is consistently \( +2 \), which characterizes it as a linear or arithmetic sequence.
Understanding that the sequence type is linear allows us to predict the pattern and derive an explicit formula quickly. Linear sequences can appear in many real-world applications such as financial predictions or resource planning, making it important to recognize and understand their characteristics.
By identifying the constant difference in our sequence as \( +2 \), a student can conceptually connect this pattern with the explicit formula \( T(n) = 2 + 2n \), where \( 2n \) reflects the consistent growth rate and \( 2 \) is the starting point or the initial term compounded by the increment.
Understanding that the sequence type is linear allows us to predict the pattern and derive an explicit formula quickly. Linear sequences can appear in many real-world applications such as financial predictions or resource planning, making it important to recognize and understand their characteristics.
By identifying the constant difference in our sequence as \( +2 \), a student can conceptually connect this pattern with the explicit formula \( T(n) = 2 + 2n \), where \( 2n \) reflects the consistent growth rate and \( 2 \) is the starting point or the initial term compounded by the increment.
Initial Condition
The initial condition provides the starting point or first term of a sequence. In recursive sequences, knowing the initial condition is crucial because it acts as the base case from which all subsequent terms are calculated. In our problem, the initial condition is given as \( T(0) = 2 \).
The initial condition \( T(0) = 2 \) sets the foundation for deriving the entire sequence. It's like the anchor point from which we execute the recursion \( T(n) = T(n-1) + 2 \). If this initial condition were different, our entire sequence—and consequently, our explicit formula—would change.
Initial conditions are often presented in mathematical problems to ensure the sequence begins at a specific point, confirming consistency with the given theorem or problem statements. Understanding the importance of initial conditions helps in maximizing the accuracy when deriving or working with recursive formulas.
The initial condition \( T(0) = 2 \) sets the foundation for deriving the entire sequence. It's like the anchor point from which we execute the recursion \( T(n) = T(n-1) + 2 \). If this initial condition were different, our entire sequence—and consequently, our explicit formula—would change.
Initial conditions are often presented in mathematical problems to ensure the sequence begins at a specific point, confirming consistency with the given theorem or problem statements. Understanding the importance of initial conditions helps in maximizing the accuracy when deriving or working with recursive formulas.
