Please help solve questions 4 and 5. Some Pythagorean Triples.Recall that a Pythagorean triple, or simply a triple, is an integral solution to the equationa² + b² = c?(1)For example, (3, 4, 5), (5, 12, 13), and (8, 15, 17) are triples. A triple is called primitiveif (a, b) = 1.Notice that two of the triples above have the property that the hypotenuse is onegreater than the second largest side. We'll find all primitive solutions to equation (1)for triangles of this form, i.e. all primitive solutions to:a² + b? = (b+ 1)²(2)1. Suppose that a and b satisfy equation (2). Expand the right hand side of equation(2) and express a² in terms of b. Conclude that a is an odd integer.2. Now let a = 2k + 1 be any odd integer. Show it's the smallest side of a righttriangle satisfying (2), i.e.a² + b? = (b+ 1)²by expressing b in terms of k.3. Check that your expression for b in terms of k is relatively prime to a = 2k + 1.4. It follows that all primitive solutions to equation (2) are of the form(2k + 1, expression for b, expression for b+1) k = 1, 2, 3, 4, ...Write down the triples for k = 1, 2, 3, 4.5. Write down all Pythagorean triples (not necessarily primitive) solving (2) withhypotenuse smaller than 50.All solutions to the special triples were known to Pythagoras. The general solutionto (1) is in the Elements. Read what's posted on the website about the Plimpton Librarytablet.

4. It follows that all primitive solutions to equation (2) are of the form (2k + 1, expression for b, expression for b+ 1) k = 1, 2, 3, 4, ... Write down the triples for k = 1, 2, 3, 4. 5. Write down all Pythagorean triples (not necessarily primitive) solving (2) with hypotenuse smaller than 50.

4. It follows that all primitive solutions to equation (2) are of the form (2k + 1, expression for b, expression for b+ 1) k = 1, 2, 3, 4, ... Write down the triples for k = 1, 2, 3, 4. 5. Write down all Pythagorean triples (not necessarily primitive) solving (2) with hypotenuse smaller than 50.

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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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

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Please help solve questions 4 and 5.

Some Pythagorean Triples.
Recall that a Pythagorean triple, or simply a triple, is an integral solution to the equation
a² + b² = c?
(1)
For example, (3, 4, 5), (5, 12, 13), and (8, 15, 17) are triples. A triple is called primitive
if (a, b) = 1.
Notice that two of the triples above have the property that the hypotenuse is one
greater than the second largest side. We'll find all primitive solutions to equation (1)
for triangles of this form, i.e. all primitive solutions to:
a² + b? = (b+ 1)²
(2)
1. Suppose that a and b satisfy equation (2). Expand the right hand side of equation
(2) and express a² in terms of b. Conclude that a is an odd integer.
2. Now let a = 2k + 1 be any odd integer. Show it's the smallest side of a right
triangle satisfying (2), i.e.
a² + b? = (b+ 1)²
by expressing b in terms of k.
3. Check that your expression for b in terms of k is relatively prime to a = 2k + 1.
4. It follows that all primitive solutions to equation (2) are of the form
(2k + 1, expression for b, expression for b+1) k = 1, 2, 3, 4, ...
Write down the triples for k = 1, 2, 3, 4.
5. Write down all Pythagorean triples (not necessarily primitive) solving (2) with
hypotenuse smaller than 50.
All solutions to the special triples were known to Pythagoras. The general solution
to (1) is in the Elements. Read what's posted on the website about the Plimpton Library
tablet.

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