How do you solve the recurrence relation P(n) = 1 + 5n by induction?

In summary, a simple recurrence relation is a mathematical relationship between a sequence of numbers where each term is defined in terms of the previous term(s). It is different from a regular recurrence relation in that it only involves one previous term. Some real-world applications include modeling natural phenomena and use in computer algorithms. To solve a simple recurrence relation, techniques such as substitution and iteration can be used. Multiple solutions are possible, but it is important to check the initial conditions and ensure they satisfy all terms in the sequence.
  • #1
Fernando Revilla
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I quote a question from Yahoo! Answers

Solve P(n) = 1 + 5n by induction?
Closed form solution: P(n) = 1 + 5n
from, P(n) = {1 if n = 1
P(n-1) + 5 if n > 1}

I have given a link to the topic there so the OP can see my response.
 
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  • #2
The recurrence relation is $p(n)=p(n-1)+5,\; p(1)=1$. Then, $p(n)=1+5n$ is a solution.

Basis Step $p(1)=1+5\cdot 0=1$.

Induction Step Suppose the relation is true for $n$. Then, $p(n)=1+5n$, so
$$p(n+1)=1+5(n+1)=1+5n+5=p(n)+5$$

As a consqeuence, the relation is true for $n+1$.
 

FAQ: How do you solve the recurrence relation P(n) = 1 + 5n by induction?

What is a simple recurrence relation?

A simple recurrence relation is a mathematical relationship between a sequence of numbers where each term in the sequence is defined in terms of the previous term(s). This creates a recursive formula that can be used to find any term in the sequence.

How is a simple recurrence relation different from a regular recurrence relation?

A simple recurrence relation only involves one previous term in the sequence, while a regular recurrence relation may involve multiple previous terms. This makes simple recurrence relations easier to solve and understand.

What are some real world applications of simple recurrence relations?

Simple recurrence relations can be used to model and predict natural phenomena, such as population growth or the spread of diseases. They are also commonly used in computer algorithms and data structures.

How do you solve a simple recurrence relation?

To solve a simple recurrence relation, you can use techniques such as substitution, iteration, or solving for a closed form solution. It is important to identify the initial conditions and base cases in order to find a general solution.

Can a simple recurrence relation have multiple solutions?

Yes, a simple recurrence relation can have multiple solutions. This is because there may be different ways to define the relationship between terms in a sequence. It is important to check the initial conditions and make sure the solution satisfies all of the terms in the sequence.

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