Katedra matematiky - Katolícka univerzita v Ružomberku
Katedra matematiky - Katolícka univerzita v Ružomberku Katedra matematiky - Katolícka univerzita v Ružomberku
66 Jan Kopka, Leonard Frobisher, George FeissnerProblem: Investigate the number of different ways a given positive integercan be written the sum of consecutive positive integers.Here is our investigation:Again we can start with the process of experimentation. We can also use ourmini-theory, which was created above. But first several specific questions.Question: Which numbers have no representation as the sum of consecutiveintegers?Answer: Powers of 2.Question: Which numbers have a unique representation?Answer: Those with only one odd factor other than 1, namely primes andproducts of a power of 2 and a prime, such as 56 = 8·7 = 2 3 ·7.Question: Which numbers have exactly two such representations?Answer: Those with exactly two distinct odd factors other than 1, such as9 = 3 2 . The two distinct factors are 3 and 9.Question: Which numbers have exactly three such representations?Answer: Those with exactly three distinct odd factors other than 1, suchas 42 = 2·3·7. The three distinct odd factors are 3,7, and 21.We are now in a position to offer a further conjecture.Conjecture 4: Let n be positive integer. The number of representationsof n as a consecutive sum is the same as the number of odd factors of theinteger n.We are pretty certain that the conjecture is true, but can we prove it?Proof: For every odd factor 2k+1 of n we can construct sum (2) and aftercounteract we always get a sum of consecutive positive integers.For example: Number 30 = 2.3.5 has three odd factors: 3, 5 and 3.5 = 15.Sum of 3 consecutive integers: 30 = 9+10+11Sum of 5 consecutive integers: 30 = 4+5+6+7+8Sum of 15 consecutive integers (or 4 consecutive positive integers):30 = (−5)+(−4)+(−3)+(−2)+(−1)+0+1+2+3+4+5+6+7+8+9=6+7+8+9.The conjecture 4 can now be claimed to be a theorem 5.We end with two remarks of a didactic nature:Remark 1: We can use a geometric approach to introduce younger studentsto this problem with the help of figures such as these:
Conjecturing: An Investigation Involving Positive Integers 67Diagrammatising numbers in geometric shapes helps students visualize patternsin particular numbers. The shapes could be said to represent an assemblyof pipes or logs of wood. The numbers below the figures show thenumber of pipes in a shape and we can refer to numbers as ’pipe’ numbers.So we investigate which numbers are pipe numbers. Younger students canalso be introduced to experimentation in mathematics by supplying themwith a collection of metal or plastic disks and having them "construct" pipenumbers.A more algebraic view:It is easy to derive the formula 1+2+3+...+n = n(n+1)/2. There areboth algebraic and geometric ways to do so. Now, a "sum" number such as5+6+7+8+9+10 can be looked at as(1+2+3+...+9+10)−(1+2+3+4) = (10·11)/2−(4·5)/2.Thus, a "sum" number is one that can be written either in the form n(n+1)/2 or n(n+1)/2−m(m+1)/2 where in the second case, m+1 < n.See how this relates to the figures: the first formula (where n = 2,3,4,5,... )produces triangles, more precisely, the triangular numbers: 3,6,10,15,...The second formula (where n = 2,3,4,5... , m = 1,2,3,4,... and m+1
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Conjecturing: An Investigation Involving Positive Integers 67Diagrammatising numbers in geometric shapes helps students visualize patternsin particular numbers. The shapes could be said to represent an assemblyof pipes or logs of wood. The numbers below the figures show thenumber of pipes in a shape and we can refer to numbers as ’pipe’ numbers.So we investigate which numbers are pipe numbers. Younger students canalso be introduced to experimentation in mathematics by supplying themwith a collection of metal or plastic disks and having them "construct" pipenumbers.A more algebraic view:It is easy to derive the formula 1+2+3+...+n = n(n+1)/2. There areboth algebraic and geometric ways to do so. Now, a "sum" number such as5+6+7+8+9+10 can be looked at as(1+2+3+...+9+10)−(1+2+3+4) = (10·11)/2−(4·5)/2.Thus, a "sum" number is one that can be written either in the form n(n+1)/2 or n(n+1)/2−m(m+1)/2 where in the second case, m+1 < n.See how this relates to the figures: the first formula (where n = 2,3,4,5,... )produces triangles, more precisely, the triangular numbers: 3,6,10,15,...The second formula (where n = 2,3,4,5... , m = 1,2,3,4,... and m+1
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