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A drawing of diagonals for four different polygons is listed below. What is the number of diagonals for a 100-sided polygon?

  • math -

    100 sided polygon has = n(n-3)/2 diagonals

    4850 diagonals

  • math -

    The number of diagonals in the first series of polygons are

    Number of sides...........n = 3....4....5....6....7....8
    Number of diagonals.....N = 0....2....5....9...14..20
    1st Difference.......................2....3....4....5....6
    2nd Difference.........................1....1....1....1

    We therefore, have a finite difference sequence with the 2nd differences constant at 1. This means that the general expression for the number of diagonals in any n-gon is of the form N = an^2 + bn + c.

    Using the data, we can write
    a(3^2) + b(3) + c = 0 or 9a + 3b + c = 0
    a(4^2) + b(4) + c = 2 or 16a + 4b + c = 2
    a(5^2) + b(5) + c = 5 or 25a + 5b + c = 5

    Solving this set of equations leads us to a = 1/2, b = -3/2, and c = 0 resulting in N = n^2/2 - 3n/2 = n(n - 3)/2.

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