91Ó°ÊÓ

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.

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 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.)

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 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?