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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Proportion
Percentages
Probability, statistics
Supplementary material
Open sentences
Spatial symmetry
Sequences
Index of new concepts
Sequences (supplementary material)
Starting from number \latex{ 1 }, the neighbouring houses in a street form a sequence. The terms of a sequence can be numbers, plane figures, solids, etc.
Example 1
Connect the midpoints of an equilateral triangle and colour the resulting triangle in the middle. Repeat the same with the remaining yellow triangles in each step.
  1. In your notebook, construct the third term of the sequence.
  2. In each large triangle, determine the sum of the perimeters of the coloured triangles if the side of the large triangle is \latex{ 8 } \latex{ cm } long.
  3. What part of the area of each large triangle is coloured?
If the procedure described in the example is repeated numerous times, the result is a Sierpiński triangle. This triangle is a fractal characterised by self-similarity.
\latex{ 4 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
Wacław Sierpiński was a Polish mathematician. Look up when he lived. Find images of other fractals.
Solution
  1. The midpoint of a side can be determined by constructing its perpendicular bisector. Using this method, you can construct the third triangle seen in the image.
  2. In the first triangle, there is no coloured triangle, so its perimeter is \latex{ 0 }.
    In the second triangle, the side of the coloured triangle is \latex{8\div 2=4} \latex{ cm } long, so its perimeter is \latex{3\times 4=12} \latex{ cm } long. 

In the third triangle, the side of a smaller coloured triangle is \latex{4\div 2=2} \latex{ cm } long, so its perimeter is \latex{3\times 2=6} \latex{ cm } long. 
In total, the perimeter of the larger and the three smaller coloured triangles in the third image is
\latex{12+3\times 6=30\;cm} long.
 
In the fourth triangle, the side of the smallest coloured triangle is \latex{2\div 2=1} \latex{ cm } long, so its perimeter is
\latex{3\times 1=3} \latex{ cm }.
In total, the perimeter of the largest, the three smaller and the nine smallest coloured triangles in the fourth image is:
\latex{12+3\times 6+9\times 3=57} \latex{ cm }. 
The four perimeters form the following sequence: \latex{0}; \latex{12}; \latex{30}; \latex{57}; ...
  1. In the first triangle, there is no coloured triangle, so its area is \latex{ 0 }.
    The second triangle was divided into \latex{ 4 } congruent smaller triangles. One of these is coloured, meaning that the \latex{\frac{1}{4}} part of the large triangle is coloured. 
In the third triangle, connect the midpoints of the larger coloured triangle's sides, dividing the original large triangle into \latex{ 16 } small triangles. \latex{4 + 3 = 7} of these small traingles are coloured, meaning that the \latex{\frac{7}{16}} part of the large triangle is coloured.
Using the same method, the fourth triangle can be divided into \latex{ 64 } small triangles. \latex{16+3\times 4+9=47} of these small triangles are coloured, meaning that \latex{\frac{47}{64}}​ part of the large triangle is coloured.
The resulting fractions form the following sequence: \latex{0} ; \latex{\frac{1}{4}} ; \latex{\frac{7}{16}}; \latex{\frac{47}{64}}; ...
Example 2
Continue the sequence \latex{ 2; } \latex{ 5; } \latex{ 8; } \latex{ 11; } ... with five consecutive terms. Represent the sequence in a coordinate system.
Solution
It can be observed that the difference between consecutive terms is \latex{ 3 }, thus the following terms are: \latex{ 14; } \latex{ 17; } \latex{ 19; } \latex{ 22; } \latex{ 25 }. This is a list of numbers that produce a remainder of \latex{ 2 } when divided by \latex{ 3 }.
In a coordinate system, there is a corresponding point for every term of the sequence. The horizontal coordinate is the number of the term, while the vertical coordinate is the term of the sequence.
It can be observed that the points form a straight line.
\latex{ 26 }
\latex{ 23 }
\latex{ 20}
\latex{ 17}
\latex{ 14}
\latex{ 11}
\latex{ 8}
\latex{ 5}
\latex{ 2}
\latex{ 1}
\latex{ 2}
\latex{ 3}
\latex{ 4}
\latex{ 5}
\latex{ 6}
\latex{ 7}
\latex{ 8}
\latex{ 9}
\latex{ 10}
The sequence could also be continued in a different way, for example:  \latex{ 2; } \latex{ 5; } \latex{ 8; } \latex{ 11; } \latex{ 8; } \latex{ 5; } \latex{ 2; } \latex{ 5; } \latex{ 8; } \latex{ 11; } ...
The continuation of a sequence is certain only if the rule of the sequence is determined.
\latex{ 11 }
\latex{ 8 }
\latex{ 5 }
\latex{ 2 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
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Exercises
{{exercise_number}}. Determine the first \latex{ 10 } terms of each sequence.
  1. The natural numbers that produce a remainder of \latex{ 3 } if divided by \latex{ 5 } in ascending order, starting from \latex{ 3 }.
  2. The natural numbers in binary form in ascending order, starting from \latex{ 0 }.
  3. The prime numbers in ascending order, starting from \latex{ 2 }.
  4. Fractions with \latex{ 1 } as a numerator in descending order, starting from \latex{ 1 }.
{{exercise_number}}. Construct sequences using logical blocks so that the consecutive terms differ from each other
  1. in one property;
  2. in two properties;
  3. in three properties.
{{exercise_number}}. List the terms of the sequences if a term can be calculated by adding the digits of the previous term. Start the sequences with the following terms.
  1. \latex{ 864 }
  1. \latex{ 7,696 }
  1. \latex{ 468,739 }
  1. \latex{ 348,725,638, 103 }
{{exercise_number}}. In the following sequence, a term can be obtained by taking its preceding term and subtracting the term before it: \latex{\triangle;\bigcirc;\Box;} \latex{3;4;1;-3};..
  1. Continue the sequence with six terms.
  2. With what numbers can the symbols be replaced?
{{exercise_number}}. In the following sequence, every term is the average of its two preceding terms: \latex{\heartsuit ;\diamondsuit ;\clubsuit ;} \latex{4;2};...
  1. Continue the sequence with six terms.
  2. With what numbers can the symbols be replaced?
{{exercise_number}}. Sam builds towers by stacking dice one on top of the other so that the visible surface of the towers contains the least number of dots. The consecutive towers consist of \latex{ 1; 2; 3; 4; } ... dice.
  1. What is the surface area of each consecutive tower if the edges of the dice measure \latex{ 1 } \latex{ cm ?}
  2. How many dots in total are visible on each consecutive tower?
{{exercise_number}}. Lucy is colouring the natural numbers in order. She colours the first number blue, the following two numbers red, the following three green, then she starts again with a blue number and so on. What colour will the following numbers be?
  1. \latex{ 2,007 }
  1. \latex{ 13,862 }
  1. \latex{ 865,000 }
{{exercise_number}}. Determine the rules and continue the sequences.
  1. \latex{333};  \latex{33.3};  \latex{3.33};  \latex{0.333}; ...
  2. \latex{0.3};  \latex{0.33};  \latex{0.333};  \latex{0.3333}; ...
  3. \latex{0};  \latex{1};  \latex{2};  \latex{10};  \latex{11};  \latex{12};  \latex{20};  \latex{21};  \latex{22};  \latex{100};  \latex{101}; ...
Quiz
Can you determine the rule of the following sequence?
\latex{ 8; 5; 4; 9; 1; 7; 6; 3; 2; 0 }
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