Please use neat handwriting and explain what is happening For the following you must prore that allnumbers n2d0Can be repeesentedFor non negahive i and j by using Strong indectran.for

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Description: Please use neat handwriting and explain what is happening For the following you must prore that allnumbers n2d0Can be repeesentedFor non negahive i and j by using Strong indectran.for

To prove: All numbers n≥20 can be represented as n=4·i+5·j for non-negative i and j by strong…

Answered: For the following you must prore that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please use neat handwriting and explain what is happening

For the following you must prore that all
numbers n2d0
Can be repeesen
ted
For non negahive i and j by using Strong indectran.
for

Expert Solution

Solution:

To prove: All numbers $n \geq 20$ can be represented as $n = 4 \cdot i + 5 \cdot j$ for non-negative i and j by strong induction.

Proof:

Let $P (n)$ be the statement such that $P (n) = 4 \cdot i + 5 \cdot j$ for non-negative i and j .

Perform the basis step in strong induction:

Show that the statements P(20), P(21), P(22), and P(23) are true.

$P (20) = 4 \cdot 0 + 5 \cdot 4 P (21) = 4 \cdot 4 + 5 \cdot 1 P (22) = 4 \cdot 3 + 5 \cdot 2 P (23) = 4 \cdot 2 + 5 \cdot 3$

Hence, the statements P(20), P(21), P(22), and P(23) are true.

Now, any value n (20 ≤ n ≤ k) where k ≥ 23, can be expressed as $n = 4 i + 5 j$ with i and j being
non-negative integers.

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