Recursive description of a sequence. Today it’s all about the recursive description of a sequence. Have a look!!

### Recursive description of a sequence

Suppose, there is a sequence like . Here both and are constants.

Also, the sequence holds true for any value of greater than .

Now the recursive description of a sequence means to convert this sequence to a linear difference equation.

So, to convert the sequence to a linear difference equation, I’ll take a few steps.

First of all, I’ll replace with .

Therefore, the sequence will be,

Next, I’ll simplify it.

So it will look like

Now I can also rewrite it as

But I already know that .

So this means

This is the difference equation derived from the sequence.

Okay, now I’ll give some more examples of how to convert the sequence to a difference equation.

#### Examples of the recursive description of a sequence

So here are the examples.

*Disclaimer: These examples do not belong to me. I have chosen these from a book. At the end of the post, I have given the due reference.*

##### Example 1

According to Stroud and Booth (2011) “Find a recursive description corresponding to … the following prescription for the output of a sequence: where is an integer .”

##### Solution

Here the sequence is

Also, is an integer for any value greater than or equal to .

As I have mentioned above, ‘a recursive description corresponding to the prescription for the output of a sequence’ means to derive a linear difference equation from a sequence.

First of all, I’ll replace with .

Therefore the sequence will be

Next, I’ll simplify it.

So it will look like

Now I can also rewrite it as

But I already know that .

So this means

Thus I can say that the recursive description of the sequence is

And this is the answer to the given example.

Now comes my second example.

##### Example 2

According to Stroud and Booth (2011) “Find a recursive description corresponding to … the following prescription for the output of a sequence: where is an integer .”

##### Solution

Here the sequence is

Also, is an integer for any value greater than or equal to .

Again I’ll do the same as Example 1.

First of all, I’ll replace with .

Therefore the sequence will be

Next, I’ll simplify it.

So it will look like

Now I can also rewrite it as

But I already know that .

So this means

Thus I can say that the recursive description of the sequence is

And this is the answer to the given example.

Now comes my last example.

##### Example 3

According to Stroud and Booth (2011) “If

where and are constants, show that ….”

##### Solution

Here the sequences are

and

Also, here both and are constants.

And I have to prove that

So I’ll start with

###### Step 1

First of all, I’ll replace with .

So it will be .

And this means .

But I already know that .

So this means .

Also, I already know from the given example that

Thus the sequence will be

Next, I’ll prove the relation

###### Step 2

Now I’ll start from the left-hand side (LHS) of the equation.

Thus it will be

Next, I’ll simplify it further to get

So I can conclude that the relation is proved. And this is the solution to my last example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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