Question

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

Short answer

The sequence alternates: \( T(n)=1 \) for even \( n \), and \( T(n)=0 \) for odd \( n \).

Step-by-step solution

Understand the Recursive Formula

The given recursive formula is \( T(n+1) = -T(n) + 1 \). This means the value of \(T\) at position \( n+1 \) depends on the value at position \( n \). The starting condition is \( T(0) = 1 \).

Compute Initial Few Terms

Using the initial condition \( T(0) = 1 \):- For \( n=1 \), \( T(1) = -T(0) + 1 = -1 + 1 = 0 \).- For \( n=2 \), \( T(2) = -T(1) + 1 = 0 + 1 = 1 \).- For \( n=3 \), \( T(3) = -T(2) + 1 = -1 + 1 = 0 \).

Identify the Pattern

Based on calculated terms \( T(0)=1 \), \( T(1)=0 \), \( T(2)=1 \), and \( T(3)=0 \), we observe a repeating pattern: \( 1, 0, 1, 0, \ldots \). This pattern alternates between \( 1 \) and \( 0 \) for odd and even \( n \) respectively.

Conclude the General Solution

The terms in the sequence alternate such that for even \( n \), \( T(n) = 1 \), and for odd \( n \), \( T(n) = 0 \). This reflects a simple alternating pattern.

Key concepts

Recursive Formula

A **recursive formula** is a rule that defines the terms in a sequence using previous terms. In simple terms, it provides a way to generate each new term based on the ones that came before it. This is especially helpful when working with sequences where each term is not isolated but connected to the others. In our exercise, the recursive formula given is \( T(n+1) = -T(n) + 1 \). Here, every term is derived from the previous one. This hinges on knowing the term before, which makes understanding and using recursive formulas a dynamic way of exploring sequence behavior.

When analyzing recursive formulas, it's important to note:- The nature of the rule often shows the relationship between terms.- You initiate the sequence using an **initial condition**.- From this initial condition, you can apply the formula step-by-step to get further terms.

Let's think of recursive formulas like setting up a chain reaction. Once you have the first piece in place, you can determine any subsequent piece by following the rules set by the formula.

Initial Condition

The **initial condition** in a recursive sequence is the starting point or base from which all subsequent terms are calculated. It acts like the seed of the sequence, without which the sequence cannot be unfolded. For the problem at hand, the initial condition is given as \( T(0) = 1 \). This means the sequence starts with 1 at \( n=0 \).

Initial conditions are crucial because:- They provide the necessary information to start applying the recursive formula.- Without them, it's impossible to accurately generate the sequence.- They often directly impact the nature and behavior of the sequence.
Set the initial condition correctly, and you're well on your way to finding other terms. As we saw in the exercise, from \( T(0) = 1 \), we could determine \( T(1) = 0 \), \( T(2) = 1 \), and so on. This stepwise computation only became possible with that initial value.

Alternating Pattern

An **alternating pattern** in a sequence refers to a regular fluctuation between different values or states. In this context, an alternating pattern isn't just random switching but follows a recognizable, systematic rule. Our sequence demonstrates such a pattern:

- The terms calculated from the recursive formula are \( 1, 0, 1, 0, \ldots \) - At even positions (i.e., \( T(0), T(2), T(4), \ldots \)), the term is 1. - At odd positions (i.e., \( T(1), T(3), T(5), \ldots \)), the term is 0.

This consistent back-and-forth gives the sequence its alternating character. Recognizing such patterns not only helps in predicting further terms without additional computation but also offers insights into the structure of the sequence. Alternating patterns like these are common in sequences involving periodic or oscillating behaviors.