Question

Devise a recursive algorithm to find the term of the sequence defined by${\mathbf{a}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathbf{a}}_{\mathbf{1}}\mathbf{=}\mathbf{2}\mathbf{,}{\mathbf{a}}_{\mathbf{2}}\mathbf{=}\mathbf{3}\mathbf{\text{and}}{\mathbf{a}}_{\mathbf{n}}\mathbf{=}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{2}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{3}}\mathbf{\text{for}}\mathbf{n}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{\dots }$

Short answer

The appropriate algorithm is given.

Step-by-step solution

 Step 1: Recursive algorithm

Recursive algorithms call themselves "smaller (or simpler)" input values and get the output for the current input by performing straightforward operations on the output from the smaller (or simpler) input.

More generally, one can use a recursive method to solve a problem if solutions to smaller versions of the same problem can be used and the smaller versions can be reduced to situations that are simple to tackle.

A recursive algorithm, for instance, can be used to determine the members of a set that has been constructed recursively or the value of a function that has been defined in this way.

Find the appropriate algorithm

Here we have given that,

${\mathrm{a}}_0=1,{\mathrm{a}}_1=2,{\mathrm{a}}_2=3\text{and}{\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}-2}+{\mathrm{a}}_{\mathrm{n}-3}\text{for}\mathrm{n}=3,4,5,\dots$

and for.

Now, suppose the name of the algorithm as “term”

Let is a positive integer.

Procedure term ( n : positive integer)

When the integer is zero then$a_0=1$

If n = 0 then

Return 1

When the integer is 1 then$a_1=2$

If n = 1 then

Return 2

When the integer is 2 then$a_2=3$

If n = 2 then

Return 3

Using $a_n=a_{n-1}+a_{n-2}+a_{n-3}$when is at least 3.

Else return$\mathrm{term}(n-1)+\mathrm{term}(n-2)+\mathrm{term}(n-3)$ .