Question

The middle-square method for generating pseudorandom

numbers begin with an n-digit integer. This number is

squared, initial zeros are appended to ensure that the result

has 2n digits, and its middle n digits are used to form the next

number in the sequence. This process is repeated to generate

additional terms.

9. Find the first eight terms of the sequence of four-digit

pseudorandom numbers generated by the middle square

method starting with 2357.

Short answer

Conclusion The first eight terms of the sequence are:

$\mathbf{2357}\mathbf{,}\mathbf{5554}\mathbf{,}\mathbf{8469}\mathbf{,}\mathbf{7239}\mathbf{,}\mathbf{4031}\mathbf{,}\mathbf{2489}\mathbf{,}\mathbf{1951}\mathbf{,}\mathbf{8064}$

Step-by-step solution

Step 1

Given

${\mathit{x}}_{\mathbf{0}}\mathbf{=}\mathbf{2357}$

The initial number contains$\mathbf{4}$digits

Step1: Square the number and add initials zeros until the number contains$\mathbf{8}$digits

${\mathbf{2357}}^{\mathbf{2}}\mathbf{=}\mathbf{5555449}\mathbf{=}\mathbf{05555449}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{5554}$

Step 2

Step2: Square the number and add initials zeros until the number contains $\mathbf{8}$digits

${\mathbf{5554}}^{\mathbf{2}}\mathbf{=}\mathbf{30846916}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

$\mathbf{\times }\mathbf{2}\mathbf{=}\mathbf{8469}$

Step 3

Step3: Square the number and add initials zeros until the number contains$\mathbf{8}$digits

${\mathbf{8469}}^{\mathbf{2}}\mathbf{=}\mathbf{71723961}$

The next number in the sequence is then the middlerole="math" localid="1668672291450" $\mathbf{4}$ digits of the previous number

$\mathit{x}\mathbf{3}\mathbf{=}\mathbf{8469}$

Step 4

Step4: Square the number and add initials zeros until the number contains$\mathbf{8}$digits

${\mathbf{7239}}^{\mathbf{2}}\mathbf{=}\mathbf{52403121}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

$\mathit{x}\mathbf{4}\mathbf{=}\mathbf{4031}$

Step 5

Step5: Square the number and add initials zeros until the number contains $\mathbf{8}$digits

${\mathbf{4031}}^{\mathbf{2}}\mathbf{=}\mathbf{16248961}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

$\mathbf{\times }\mathbf{5}\mathbf{=}\mathbf{2489}$

Step 6

Step6: Square the number and add initials zeros until the number contains $\mathbf{8}$digits

${\mathbf{2498}}^{\mathbf{2}}\mathbf{=}\mathbf{6195121}\mathbf{=}\mathbf{06195121}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

$\mathbf{\times }\mathbf{6}\mathbf{=}\mathbf{1951}$

Step 7

Step7: Square the number and add initials zeros until the number contains$\mathbf{8}$digits

${\mathbf{1951}}^{\mathbf{2}}\mathbf{=}\mathbf{3806401}\mathbf{=}\mathbf{03806401}$

The next number in the sequence is then the middle $\mathbf{4}$digits of the previous number

$\mathit{x}\mathbf{7}\mathbf{=}\mathbf{8064}$