Problem K. 518. (November 2016)
K. 518. The numbers in the figures and further figures created in the same way are called hexagonal numbers. Without proof, find a formula for the \(\displaystyle n\)th hexagonal number, and use it to show that 2016 is a hexagonal number.
(6 pont)
Deadline expired on December 12, 2016.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. „Emeletenként” nézve: \(\displaystyle 1 = 1 \cdot 1\), \(\displaystyle 6 = 3 \cdot 2\), \(\displaystyle 15 = 5 \cdot 3\), \(\displaystyle 28 = 7 \cdot 4\), az ötödik hatszögszám \(\displaystyle 9 \cdot 5 = 45\), a hatodik pedig \(\displaystyle 11 \cdot 6 = 66\) és így tovább. Az n. hatszögszám \(\displaystyle (2n – 1) \cdot n\). Mivel \(\displaystyle 2016 = 63 \cdot 32\), így a 2016 a 32. hatszögszám.
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101 students sent a solution. 6 points: 84 students. 4 points: 6 students. 3 points: 2 students. 2 points: 5 students. 1 point: 1 student. 0 point: 3 students.
Problems in Mathematics of KöMaL, November 2016