Chapter 6

Devise a formula for the \(n\)th pentagonal number and for the \(n\)th hexagonal number.

Derive an algebraic formula for the pyramidal numbers with triangular base and one for the pyramidal numbers with square base.

Show that in a harmonic proportion the sum of the extremes multiplied by the mean is twice the product of the extremes.

Nicomachus defined a subcontrary proportion, which occurs when in three terms the greatest is to the smallest as the difference of the smaller terms is to the difference of the greater. Show that \(3,5,6\), are in the subcontrary proportion. Find two other sets of three terms that are in subcontrary proportion.

Nicomachus defined a "fifth proportion" to exist whenever among three terms the middle term is to the lesser as their difference is to the difference between the greater and the mean. Show that \(2,4,5\), are in fifth proportion. Find two more triples in this proportion.

Solve Diophantus's Problem I-27 by the method of I-28: To find two numbers such that their sum and product are given. Diophantus gives the sum as 20 and the product as \(96 .\)

Solve Diophantus's Problem II-10: To find two square numbers having a given difference. Diophantus puts the given difference as 60 . Also, give a general rule for solving this problem given any difference.

Solve Diophantus's Problem II-13 by the method of the double equation: From the same (required) number to subtract two given numbers so as to make both remainders square. (Take 6,7 , for the given numbers. Then solve \(x-\) \(\left.6=u^{2}, x-7=v^{2} .\right)\)

Solve Diophantus's Problem B-9: To divide a given number into two parts such that the sum of their cubes is a given multiple of the square of their difference. (The equations become \(x+y=a, x^{3}+y^{3}=b(x-y)^{2}\). Diophantus takes \(a=20\) and \(b=140\) and notes that the necessary condition. for a solution is that \(a^{3}\left(b-\frac{3}{4} a\right)\) is a square.)

Solve Diophantus's Problem IV-9: To add the same number to a cube and its side and make the second sum the cube of the first. (The equation is \(x+y=\left(x^{3}+y\right)^{3}\). Diophantus begins by assuming that \(x=2 z\) and \(y=27 z^{3}-2 z\).)

Solve Diophantus's Problem \(\mathrm{V}-10\) for the two given numbers 3,9

Book VI of the Arithmetica deals with Pythagorean triples. For example, solve Problem VI-16: To find a right triangle with integral sides such that the length of the bisector of an acute angle is also an integer. (Hint: Use Elements VI-3, that the bisector of an angle of a triangle cuts the opposite side into segments in the same ratio as that of the remaining sides.)

Provide the analysis for Elements XIII-4: If a straight line is cut in extreme and mean ratio, the sum of the squares on the whole and on the lesser segment is triple the square on. the greater segment.

Show that a regular hexagon of given perimeter has a greater area than a square of the same perimeter.

Find the volume of a torus by applying Pappus's theorem. Assume that the torus is formed by revolving the disk of radius \(r\) around an axis whose distance from the center of the disk is \(R>r\).

Solve Epigram 116: Mother, why do you pursue me with blows on account of the walnuts? Pretty girls divided them all among themselves. For Melission took two-sevenths of them from me, and Titane took the twelfth. Playful Astyoche and Philinna have the sixth and third. Thetis seized and carried off twenty, and Thisbe twelve, and look there at Glauce smiling sweetly with eleven in her hand. This one nut is all that is left to me. How many nuts were there originally? \(?^{22}\)

Solve Epigram 130: Of the four spouts, one filled the whole tank in a day, the second in two days, the third in three days, and the fourth in four days. What time will all four take to fill it?

Solve Epigram 145: A. Give me ten coins and I have three times as many as you. B. And if I get the same from you, I have five times as much as you? How many coins does each have?