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21
28
36
45
55
66
78
I   91
II
     e e
        I   105




e
Derivation
•   Two lines cut in 1 point.
•    A third line will cut the other two lines in 2 more points, giving 1 + 2 = 3
    points.
•   A fourth line will cut the other 3 lines in 3 more points, giving 3 + 3 = 6
    points.
•   So the series goes n = 1, 2, 3, 4, 5, 6, .......Number of points              0,
    1, 3, 6, 10, 15, .......The gap increases by 1 each time.
•    This is the sequence of triangular numbers which has the nth term given by
    n(n-1)/2.
                                            OR
•   If you made up a difference table, the second differences would be constant,
    so the nth term is a quadratic in n.
•   If you assume f(n) = an^2 + bn + c, where f(n) is the nth term n = 1 a +
    b + c = 0 n = 2 4a + 2b + c = 1 n = 3 9a + 3b + c = 3 and solving these
    3 equations for a, b, c, a = 1/2, b = -1/2, c = 0
•   So f (n) = n^2/2 - n/2 = n(n-1)/2 as given above. So with n lines there are n
    (n-1)/2 intersection points.

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Intersection of n lines

  • 1. 21
  • 2. 28
  • 3. 36
  • 4. 45
  • 5. 55
  • 6. 66
  • 7. 78
  • 8. I 91
  • 9. II e e I 105 e
  • 10. Derivation • Two lines cut in 1 point. • A third line will cut the other two lines in 2 more points, giving 1 + 2 = 3 points. • A fourth line will cut the other 3 lines in 3 more points, giving 3 + 3 = 6 points. • So the series goes n = 1, 2, 3, 4, 5, 6, .......Number of points 0, 1, 3, 6, 10, 15, .......The gap increases by 1 each time. • This is the sequence of triangular numbers which has the nth term given by n(n-1)/2. OR • If you made up a difference table, the second differences would be constant, so the nth term is a quadratic in n. • If you assume f(n) = an^2 + bn + c, where f(n) is the nth term n = 1 a + b + c = 0 n = 2 4a + 2b + c = 1 n = 3 9a + 3b + c = 3 and solving these 3 equations for a, b, c, a = 1/2, b = -1/2, c = 0 • So f (n) = n^2/2 - n/2 = n(n-1)/2 as given above. So with n lines there are n (n-1)/2 intersection points.

Notes de l'éditeur

  1. 21
  2. 21
  3. 21
  4. 21
  5. 21
  6. 21
  7. 21
  8. 21
  9. 21