# Solve Recurrent Equation - Step-by-Step Guide

• vtakhist
In summary, a recurrent equation is one that uses the value of the previous term in a sequence to find the value of the next term. Solving recurrent equations is important in various fields and can be done by identifying the pattern, writing out the first few terms, creating a general formula, using algebra to solve, and checking the solution. There are also tips and shortcuts that can make solving recurrent equations easier, such as looking for patterns and practicing different types of equations.

#### vtakhist

Good Day. I need to solve reccurent sequence.
P{X(n)=m}=SUM(from 1, to n-1)(p(n,k)*P{X(k)=m-1}
P{X(0}=0}=1 P{X(1)=0}=1
p(n,k)-some const witch depends on n and k.
I need full solvation of this problem.
Thanks for help.

WHAT is p{X(n)=m} ? apart from other things

Probability, that random variable X(n) will take a value m.

## 1. What is a recurrent equation?

A recurrent equation is an equation that uses the value of the previous term in the sequence to find the value of the next term. This creates a repeating pattern or series of numbers.

## 2. Why is it important to solve recurrent equations?

Solving recurrent equations is important in various fields of science and mathematics. It allows us to model and predict patterns and behaviors in natural phenomenon, as well as make calculations and predictions in financial and engineering applications.

## 3. What are the steps to solve a recurrent equation?

The steps to solve a recurrent equation are:

• Identify the pattern or sequence in the equation
• Write out the first few terms of the sequence
• Create a general formula for the sequence
• Use algebra to rearrange the formula and solve for the unknown variable
• Check your solution by plugging it back into the original equation

## 4. Can you provide an example of solving a recurrent equation step-by-step?

Yes, for example, let's solve the equation: xn+1 = 2xn + 3, with the initial term x0 = 1.

• Identify the pattern: each term is twice the previous term plus 3
• Write out the first few terms: x0 = 1, x1 = 2x0 + 3 = 5, x2 = 2x1 + 3 = 13
• Create a general formula: xn = 2xn-1 + 3
• Use algebra to solve: xn - 2xn-1 = 3 → xn - 2xn-1 - 6 = 0 → (xn - 6) - 2(xn-1 - 3) = 0
• Let y = xn - 6, then the equation becomes y - 2(y - 3) = 0 → y - 2y + 6 = 0 → y = -6
• Substitute back to solve for x: xn - 6 = -6 → xn = 0
• Check solution: xn+1 = 2(0) + 3 = 3, which is the next term in the sequence

## 5. Are there any tips or shortcuts to solving recurrent equations?

Yes, there are a few tips that can make solving recurrent equations easier:

• Look for a pattern or sequence in the equation
• Write out the first few terms to help identify the pattern
• Use algebra to simplify and rearrange the equation
• Always check your solution by plugging it back into the original equation
• Practice solving different types of recurrent equations to become more familiar with the process