Recurrence Relations Solve Online

A recurrence relation is an equation with the intention of recursively describes a series: each term of the series is defined as a function of the above terms. The difference equations refer to a particular type of recurrence relation. Note but the "difference equation" is commonly used to refer to any recurrence relation. Online is the collection of resources. The online is provided by the free examples and the free assignments questions with answers.

An example of a recurrence relation is the logistic map:

x n + 1 = r x n (1-xn)

Recurrence relations solve online – Examples:

Recurrence relations solve online - Example 1:

Let  .

`x_n := x_{n-1}*x_{n-2}`            if n > 1
`x_1`                        if n = 1
`x_0`                        if n = 0
0                         if n < 0

Show that `x_n = x_{1}^{F_{n-1}}*x_{2}^{F_{n-2}} \!` where `F_n\!` is the n-th Fibonacci number   (F0 = F1 = 1 and F(n < 0) = 0)

Solution:

As before: induction is da way!

BASIS: For n = 2! we have `x_2 = x_1^{F_1}*x_0^{F_{0}} = x_1*x_0 \!`

INDUCTIVE STEP: `x_{n-2} = x_1^{F_{n-3}}*x_0^{F_{n-4}} \! and x_{n-1} = x_1^{F_{n-2}}*x_0^{F_{n-3}} \!` and

so `x_n = x_{n-1}*x_{n-2} = (x_1^{F_{n-2}}*x_0^{F_{n-3}}) * (x_1^{F_{n-3}}*x_0^{F_{n-4}}) `

= `x_1^{F_{n-2}+F_{n-3}}*x_0^{F_{n-3}+F_{n-4}}`

= `x_{1}^{F_{n-1}}*x_{2}^{F_{n-2}} !`

Recurrence relations solve online – More Problems:

Recurrence relations tutor - Example 1:

Let

`x_n = x_(n-1) + x_(n-2)`     if n > 1

`x_(1) epsi N`                       if n = 1

`x_(0) epsi N`                       if n = 0

0                                if n < 0

Show that `x_n = x_{1}F_{n-1} + x_{0}F_{n-2} \,\! "where" F_n\!` is the n-th Fibonacci number   (F0 = F1 = 1)

Solution:

Using the principle of induction we have:

BASIS: n=`2 \Rightarrow x_2 = x_1 + x_0 = x_{1}F_{1} + x_{0}F_{0}\!`

INDUCTIVE STEP: We have `x_{n-2} = x_{1}F_{n-3} + x_{0}F_{n-4}\,\! and x_{n-1} = x_{1}F_{n-2} + x_{0}F_{n-3}\!`

By definition we have:

`x_{n} := x_{n-1} + x_{n-2} = x_{1}F_{n-2} + x_{0}F_{n-3} + x_{1}F_{n-3} + x_{0}F_{n-4}`

`= x_{1}\(F_{n-2} + F_{n-3}\) + x_{0}\(F_{n-3}+F_{n-4}\) = x_{1}F_{n-1} + x_{0}F_{n-2} \mbox{ } \!`

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Recurrence relations solve online  - Example 2:

Find the limiting ratio `lim_(n->oo) (x_{n+1})/(x_n)` , for the recurrence relation `x_n = x_{n-1}+x_{n-2}.`

Solution:

We find what the limit must be, assuming that it exists.

L = `\lim_{n\rightarrow\infty} \frac{x_{n+1}}{x_n} = \lim_{n\rightarrow\infty} \frac{x_n+x_{n-1}}{x_n} = 1 + \lim_{n\rightarrow\infty} \frac{x_{n-1}}{x_n} = 1+L^{-1}`

L = 1 + `L^{-1}`

`L^2=L+1`

`L^2-L-1=0`

L = `\frac{1 \pm \sqrt{ 5 }}{2}` , via the quadratic formula.