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In my coursework i have got 7 formulas, as shown below:
triangular: ½ n2 + ½ n
pentagonal: (2.5n2 – 2.5n) + 1
hexagonal: (3n2 – 3n) + 1
heptagonal: (3.5n2 – 3.5n) + 1
octagonal: (4n2 – 4n) + 1
nonaonal: (4.5n2 – 4.5n) + 1
decagonal: (5n2 – 5n) + 1
it then tells me to derive a general equation which links to all polygonal bases, making each point of how i obtained the formula clearly explained (by explaining the stages of the process), in detail.
personally i dont know how to do this, so i was wondering if you could help me.
thanks
triangular: ½ n2 + ½ n
pentagonal: (2.5n2 – 2.5n) + 1
hexagonal: (3n2 – 3n) + 1
heptagonal: (3.5n2 – 3.5n) + 1
octagonal: (4n2 – 4n) + 1
nonaonal: (4.5n2 – 4.5n) + 1
decagonal: (5n2 – 5n) + 1
it then tells me to derive a general equation which links to all polygonal bases, making each point of how i obtained the formula clearly explained (by explaining the stages of the process), in detail.
personally i dont know how to do this, so i was wondering if you could help me.
thanks
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You can split up the k-sided polygon into the dot in the middle, plus the k "spokes" going from the centre to the vertices, plus the k triangles between the spokes. (See the attached picture.) Each spoke contains (n - 1) dots. Each triangle contains
(n - 2) + (n - 3) + ... + 3 + 2 + 1 = (1/2)(n - 1)(n - 2)
dots. So the total number of dots is
1 + k(n - 1) + (1/2)k(n - 1)(n - 2)
= 1 + k(n - 1) * [1 + (1/2)(n - 2)]
= 1 + (1/2)kn(n - 1).
(n - 2) + (n - 3) + ... + 3 + 2 + 1 = (1/2)(n - 1)(n - 2)
dots. So the total number of dots is
1 + k(n - 1) + (1/2)k(n - 1)(n - 2)
= 1 + k(n - 1) * [1 + (1/2)(n - 2)]
= 1 + (1/2)kn(n - 1).
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