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By Almeida L., Damascelli L., Ge Y.

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A. Make a drawing to demonstrate that each rectangular number is double a triangular number. b. Nikomachos found a direct formula for the triangular numbers: 1 the n th triangular number ‫ ؍‬᎑᎑ 2 n(n ؉ 1), where n starts at 1. Explain this formula. 13. In problem 9, you found the 29th, 30th, and 31st triangular numbers. Use Nikomachos’s formula to check those values. 70 m 14. How many cans will fit in a triangular display against this wall? 11 levels 15 cm 16 cans The manager of a store wants to try a new way to stack cans.

42 Patterns and Figures Additional Practice You can paste two tetrahedra together as shown in the figure. 2. How many tiles are needed to completely cover this new shape? Dot Pattern: Pattern Number: 2 3 3. Study the dot pattern above and draw the pattern for n ‫ ؍‬4. You can split each pattern in a triangle and a rectangle so that the base of the triangle has as many dots as the right side of the rectangle. The following is a sketch of how the shape can be split into a triangle and a rectangle.

The numbers, 1, 3, 6, 10,.... are called triangular numbers. 1 1 ؉2 3 1 2 ؉3 6 1 2 3 ؉4 10 8. a. Describe any connections that you see between the triangular tessellations, the dot patterns, and the triangular numbers. b. What regularities do you see in each of the patterns above? 9. The 30th triangular number is 465. Using this fact, what is the 31st triangular number? What is the 29th? 32 Patterns and Figures Triangles and Triangular Numbers D Rectangular Numbers Nikomachos was interested in finding a direct formula for the triangular numbers.

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A few symmetry results for nonlinear elliptic PDE on noncompact manifolds by Almeida L., Damascelli L., Ge Y.

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