Burton: The History Burton: The History of Ma thematics: An Introduction, Sixth Edition 3. The Beginnings of Greek Ma thematics Text © The McGraw−Hill Comp anies, 2007 P y t h a g o r e a n M a t h e m a t i c s 105 3.2 Problems 1.

Plutarch (about A.D. 100) stated that if a tri angular number is multiplied by 8, and 1 is added, then the result is a square number. Prove that this is fact and illustrate it geometrically in the case of t2. Prove that the square of any odd multiple of 3 is the difference of two tri angular numbers, specifically that [3(2n + 1)] 2 = t9n+4 − t3n+1. Prove that if tn is a tri angular number, then 9tn + 1 is also tri angular. Write each of the following numbers as the sum of three or fewer tri angular numbers: (a) 56, (b) 69, (c) 185, (d) 287.

For n ≥ 1, establish the formula (2n + 1) 2 = (4tn + 1) 2 − (4tn) 2. Verify that 1225 and 41,616 are simult aneously square and tri angular numbers.

[Hint: Finding an integer n such that tn = n(n + 1) 2 = 1225 is equivalent to solving the quadratic equation n 2 + n − 2450 = 0.] 7. An oblong number counts the number of dots in a rect angular array having one more row th an it has columns; the first few of these numbers are........................................ O1 = 2 o2 = 6 o3 = 12 o4 = 20 and in general, the nth oblong number is given by on = n(n + 1). Manually Delete Snapshots Parallels there. Prove algebraically and geometrically that (a) on = 2 + 4 + 6 + + 2n. (b) Any oblong number is the sum of two equal tri angular numbers. (c) on + n 2 = t2n.

(d) on − n 2 = n. (e) n 2 + 2on + (n + 1) 2 = (2n + 1) 2.

(f) on = 1 + 2 + 3 + + n + (n + 1) + (n − 1) + (n − 2) + + 3 + 2. In 1872, Lebesgue proved that (1) every positive integer is the sum of a square number (possibly 0 2 ) and two tri angular numbers and (2) every positive integer is the sum of two square numbers and a tri angular number. Confirm these results in the cases of the integers 9, 44, 81, and 100. Display the consecutive integers 1 through n in two rows as follows: 1 2 3 n − 1 n n n − 1 n − 2 2 1 If the sum obtained by adding the n columns vertically is set equal to the sum obtained by adding the two rows horizontally, what well-known formula results?