Problem 4 Using the iterative method, pred... [FREE SOLUTION]

Chapter 5: Problem 4

Using the iterative method, predict a solution to each recurrence relation satisfying the given initial condition. $$\begin{aligned} &a_{1}=1\\\ &a_{n}=a_{n-1}+(2 n-1), n \geq 2 \end{aligned}$$

Short Answer

Expert verified

The given recurrence relation and initial condition $a_n = a_{n-1} + (2n - 1), a_1 = 1$ generate the sequence $1, 4, 9, 16 ...$, which represents the square numbers. By observing the pattern, we can predict the solution to be $a_n = n^2$ for $n \geq 1$, which satisfies both the recurrence relation and the initial condition.

Step by step solution

Identify the initial condition

The given initial condition is $a_1 = 1$. It means that the first term of the sequence is 1.

Apply the recurrence relation to find the next terms

We can use the recurrence relation $a_n = a_{n-1} + (2n - 1)$ to find the next terms of the sequence.

Find $a_2$

For $n = 2$, substitute the value of $n$ and $a_1$ into the recurrence relation: \[a_2 = a_1 + (2\cdot 2 - 1) = 1 + (4 - 1) = 1 + 3 = 4\]

Find $a_3$

For $n = 3$, substitute the value of $n$ and $a_2$ into the recurrence relation: \[a_3 = a_2 + (2\cdot 3 - 1) = 4 + (6 - 1) = 4 + 5 = 9\]

Find $a_4$

For $n = 4$, substitute the value of $n$ and $a_3$ into the recurrence relation: \[a_4 = a_3 + (2\cdot 4 - 1) = 9 + (8 - 1) = 9 + 7 = 16\]

Observe the pattern

The sequence generated by the given recurrence relation and initial condition is: \[1, 4, 9, 16, \dots\] This sequence represents the square numbers. So, we can predict that the solution to the recurrence relation is $a_n = n^2$ for $n \geq 1$. To verify this solution, we can now confirm that it fits the recurrence relation and initial condition:

Verify the solution for the initial condition

For $n = 1$, the predicted solution gives: \[a_1 = 1^2 = 1\] This matches the given initial condition $a_1 = 1$.

Verify the solution for the recurrence relation

Now, let's check if the proposed solution $a_n = n^2$ satisfies the recurrence relation $a_n = a_{n-1} + (2n - 1)$. We have: \[n^2 = (n-1)^2 + (2n - 1)\] Expand the equation: \[n^2 = n^2 - 2n + 1 + 2n - 1\] Simplify the equation: \[n^2 = n^2\] The equation holds true, confirming that the predicted solution $a_n = n^2$ for $n \geq 1$ satisfies the given recurrence relation and initial condition.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Iterative Method

The iterative method is a technique used to solve recurrence relations by repeatedly applying a formula to find successive terms in a sequence. It's like a step-by-step approach where you keep using a rule to progress from one term to the next. This method is particularly helpful in mathematical problems where a sequence is defined in terms of its previous terms, rather than an explicit formula.

You start with an initial term or condition, which is known.
Apply the recurrence relation formula to find the next few terms.
Observe the pattern and derive a general solution.

In our example, the recurrence is applied starting from the known term, and each step builds upon the previous term. By understanding this approach, you can uncover how complex sequences develop over iterations.

Initial Condition

The initial condition is the starting point of the sequence. It provides the first value that is needed to kick-start our iterations and find subsequent terms using the recurrence relation. In the context of our example, the initial condition is given as $a_1 = 1$. This means that the very first term in the sequence is 1.

Essentially, without this initial condition, there is no basis to begin the iterative process.
It acts as our anchor or foundation for generating further terms.

So always remember, knowing your initial condition is crucial as it directly affects the values of all subsequent terms derived using the recurrence relation.

Sequence Pattern

Once you have generated a couple of terms using the iterative method, it's essential to observe them to identify a pattern. Recognizing a sequence pattern is akin to noticing regularities or a rule among the numbers which can lead to formulating a general expression for the sequence.
In our example, the first several terms were $1, 4, 9, 16, \dots$. Observing these terms reveals that they are perfect squares: $1^2, 2^2, 3^2, 4^2, \dots$.

This pattern suggests that the sequence follows the rule $a_n = n^2$.
Finding such patterns is key to solving recurrence problems, as it allows us to express the sequence with a simple formula.

By identifying these patterns, you not only solve the problem at hand, but you also gain insight into the nature of such sequences in general.

Verification of Solution

After predicting a pattern or a solution, verification is necessary to ensure that it truly fits both the recurrence relation and the initial condition. This confirmation is a crucial step and involves checking if the general solution behaves correctly according to the rules of the problem.
Let's break it down:

Verify the initial condition where $a_1 = 1$. For our solution $a_n = n^2$, clearly $a_1 = 1^2 = 1$, which aligns perfectly.
Validate the recurrence relation, confirming that each term computed by $a_n = n^2$ matches when using previous terms via the relation $a_n = a_{n-1} + (2n - 1)$.

This process is like cross-checking your work鈥攅nsuring the predicted solution is not just a coincidence but a valid, mathematical rule fit for all terms defined by the initial problem. With both checks orderly satisfied, one can confidently say the sequence $a_n = n^2$ is a verified solution.

91影视

Using the iterative method, predict a solution to each recurrence relation satisfying the given initial condition. $$\begin{aligned} &a_{1}=1\\\ &a_{n}=a_{n-1}+(2 n-1), n \geq 2 \end{aligned}$$

Short Answer

Step by step solution

Identify the initial condition

Apply the recurrence relation to find the next terms

Find \(a_2\)

Find \(a_3\)

Find \(a_4\)

Observe the pattern

Verify the solution for the initial condition

Verify the solution for the recurrence relation

Key Concepts

Iterative Method

Initial Condition

Sequence Pattern

Verification of Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Applied Mathematics

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.